Exponential and Logarithmic Equations 13.5 13.4 13.3 13.2 13.1

451451_IM3_TIG_Ch13_917-988.indd 917

© Carnegie Learning and the amount of medication in a patient’s bloodstream. can be used to describe the intensity of sounds They can even be used Logarithmic equations to help solve crimes! Exponential and Logarithmic Equations 13.5 13.4 13.3 13.2 13.1

Logarithmic Equations . Logarithmic Equations Applications ofExponentialand Solving LogarithmicEquations Solving ExponentialEquations . Properties ofLogarithms Exponential andLogarithmicForms So WhenWill IUseThis? Logging On What’s Your Strategy? Mad Props All t he PiecesofthePuzzle . . . 13 971 953 941 933 919 30/11/1317/12/13 10:1412:16AMPM 917 13 451451_IM3_TIG_Ch13_917-988.indd 1 917A exponential andlogarithmicequationsincontext. derive theproperties oflogarithms. Studentsexploremethodsforsolvinglogarithmicequationsandsolve alternative expressions, includingestimating thevaluesoflogarithmsonanumberlineandthenusethisunderstandingto logarithmic equations.Studentsbeginbybuildingunderstandingandfluencywithexponential In thischapter, studentsusetheirunderstandingofexponential andlogarithmicfunctionstosolveexponential

13.2 13.3 13.1

Exponential and Logarithmic Equations Chapter 13 Exponential and Lesson Properties of Logarithmic Exponential Logarithms Equations Solving Forms F.BF.5(+) . F5+ 1 F.BF.5(+) .F5+ 2 F.BF.5(+) CSPacing CCSS F.LE.4 Chapter 13Overview 1 problems withandwithout context. solution strategiesandtoapplythem Questions askstudentstoanalyzethese sides ofanequationtosolveit. also explore takingthelogarithmofboth to solveexponentialequations.Students Formula andusethisformulaasastrategy Students derivetheChangeofBase logarithmic expressions. the properties torewrite Questions askstudentstouse same baseandargument. identify thevalueofalogarithmwith Rule, andthePowerRuleofLogarithms Property, theProduct Rule,theQuotient In thislesson,studentsderivetheZero the different unknowns. solving logarithmicequationsbyidentifying Students developdifferent strategiesfor logarithmic equations. bases, andsolvesimpleexponential of logarithmicexpressions withdifferent logarithmic equations,estimatethevalues Students convertbetweenexponentialand Highlights X X X X X Models

Worked Examples

X X Peer Analysis

Talk the Talk

Technology 17/12/13 12:16PM

© Carnegie Learning 13.5 13.4 Exponential and Lesson Applications of Logarithmic Logarithmic Equations Equations Solving F.BF.5(+) F.BF.5(+) CSPacing CCSS F.LE.4 F.LE.4 1 2 of Cooling. population growth, andNewton’s Law Students explore problems withpHlevels, involving logarithmicequations. logarithmic modelstosolveproblems In thislesson,studentsuseexponentialand different equations. efficiency ofsolutionstrategiesfor equations incontextandtoanalyzethe Questions askstudentstosolvethese previous lessons. and propertiesin theyhavelearned argument, orexponentusingthestrategies equations fordifferent unknowns—base, In thislesson,studentssolvelogarithmic Exponential and Logarithmic Equations Chapter 13 Highlights

X X X Models

Worked Examples

X Peer Analysis

Talk the Talk

Technology 917B 17/12/13 12:16PM 13 13 451451_IM3_TIG_Ch13_917-988.indd 918 918

13.5 13.4 13.3 13.2 13.1

Exponential and Logarithmic Equations Chapter 13 Exponential and Exponential and Lesson Applications of Properties of Logarithmic Logarithmic Logarithmic Exponential Logarithms Equations Equations Equations Solving Solving Forms Skills Practice Correlation forChapter13 Skills PracticeCorrelation Problem Set 9–2 Usevariouslogarithmicformulastosolveproblems 19 –24 13 –18 5–3 Determinetheappropriate unknownbaseinlogarithmicequations Estimatelogarithmstothetenthsplace 25 –30 17 –24 1–2 Solveexponentialequationsandexplainmethodsofsolving Solveexponentialequationsusingproperties oflogarithms 21 –26 11 –20 5–3 Write logarithmicexpressions asalgebraicexpressions Write logarithmicexpressions assinglelogarithms 25 –30 17 –24 2Useexponential equationsandformulastosolveproblems incontext 7 –12 6Solveforunknownsinlogarithmicequations 9 –16 9 –16 Use theChangeofBaseFormulatosolveexponentialequations 1 –10 6Useproperties oflogarithmstowriteinexpandedform 1 –16 Uselogarithmstosolvehalf-lifeproblems 1 –6 Arrangetermstocreate trueexponentialandlogarithmicequations 1 –8 Solvelogarithmicequationsandcheckanswers 1 –8 solve problems Use thelogarithmicformulaformagnitudeofanearthquaketo Vocabulary Use properties oflogarithmstosolveequationsandcheckanswers Vocabulary Objectives 17/12/13 12:16PM

© Carnegie Learning E In thislesson,youwill: lEArning gOAlS Exponential andl of thePuzzle All t • • • • • • • • • SSEntiA value ofthelogarithmisdecreasing. base isgreater than1andincreasing, the For afixedargument, whenthevalueof logarithm getslarger aswell. of theargument getslarger, thevalueof For afixedbasegreater than1,asthevalue and greater thanthenon-integer. same basewhoseargument islessthan identify theclosestlogarithmswith integer, useanumberlineasguideand To estimatealogarithmthatisnotan corresponding exponentialexpression. equal tothevalueofexponentin The valueofalogarithmicexpression is Evaluate logarithmicexpressions. number line. Estimate thevaluesoflogarithmsona logarithmic equations. Solve exponentialandsimple exponential equations. Convert logarithmicequationsinto logarithmic equations. Convert exponentialequationsinto he Pieces l id EAS ogarithmic f KEy tErm Build newfunctionsfrom existingfunctions F-BF BuildingFunctions St CO • 5. A logarithmic equation mmO involving logarithmsandexponents. use thisrelationship tosolveproblems between exponentsandlogarithms (+) Understand theinverserelationship ndA n COrE rdS fO orms r mAthEmA St AtE 13.1 tiCS 919A 17/12/13 12:16PM 13 451451_IM3_TIG_Ch13_917-988.indd 2 919B

Exponential and Logarithmic Equations Chapter 13 sometimes, neveractivityisusedtosummarizethelesson. estimate logarithmsthatare irrationalnumbersbyusinganumberlineasguide.Analways, relationship tosolveforan unknownbase,exponent,orargument inalogarithmicequation.Students Students convertbetweenexponentialandlogarithmicformsofanequation,thenusethis Overview 17/12/13 12:16PM

© Carnegie Learning Warm Up 5. 4. 3. 6. 2. 1. log of thelogarithmsinQuestions2and4. Write alogarithmicequationforlogarithmofbase2thathasvaluewhichisbetweenthevalues 2 Solve fortheunknowninexponentialequationyouwrote inQuestion3. log Convert thelogarithmicequationtoanexponentialequation. log Convert thelogarithmicequationtoanexponentialequation. log Convert thelogarithmicequationyouwrote inQuestion5toanexponentialequation. log or 2 2 Solve fortheunknowninexponentialequationyouwrote inQuestion1. x54 x53 x x 516 58 2 2 2 2 2 (8)5x (12)¯3.6 (12)¯3.5 (12)¯3.5orlog (16)5x 2 2 2 2 2 2 2 2 x 3.5 3.6 58 x ¯ 12 516 ¯ 12 2 (12)¯3.6 Exponential and Logarithmic Forms 13.1

919C 17/12/13 12:16PM 13 13 451451_IM3_TIG_Ch13_917-988.indd 4 919d

Exponential and Logarithmic Equations Chapter 13 17/12/13 12:16PM

© Carnegie Learning The graphaboveshowsthefunction log I In thislesson,youwill: LEArnIng gOALS Exponential andLogarithmicForms of thePuzzle All t into additions on this graph? What divisions can you turn intosubtractions? into additionsonthisgraph?Whatdivisionscanyouturn turn thismultiplication, turn 2 For example,what’s 4316?Ifyouknow(orcanlookup)somelogarithms, into additionanddivisionsubtraction. multiplication The advantagetousinglogarithmswasthattheycouldessentiallyturn required tocalculatewith. numbers—representing distancestootherbodiesinthesolarsystem—theywere • • • • • Astronomers especially benefited from logarithms because of the enormous Astronomers especiallybenefitedfromlogarithmsbecauseoftheenormous n thepast,logarithmshelpedmathematiciansandotherscientistssavealotoftime. Evaluate logarithmicexpressions . number line. Estimate thevaluesoflogarithmsona logarithmic equations. Solve exponentialandsimple exponential equations. Convert logarithmicequationsinto logarithmic equations. Convert exponentialequationsinto he Pieces 2 1 4 5 6 24 4 8 0 y 4 84 2 ? 2 3 4 , intoaddition:21456. 1612 16 2420 2 Exponential and Logarithmic Forms 13.1 (x).Whatothermultiplicationscanyouturn 3228 KEY TErM • 64440 36 logarithmic equation 5248 6056 5 ? x 13.1 30/11/13 10:14AM 919

17/12/13 12:16PM 919 13 13 451451_IM3_TIG_Ch13_917-988.indd 920 920

• • • • Question 1 for Share Phase, guiding Questions • • grouping vice versa. in exponentialformand expressions andequations form, giventhecorresponding and equationsinlogarithmic Students writeexpressions Problem 1

logarithmic form? before convertingitinto the exponentialrelationship Why isithelpfultoconsider logarithm orexponent? Which numberrepresents the the argument? Which numberrepresents base ofthelogarithm? Which numberrepresents the Discuss asaclass. of logarithmicequation. information anddefinition Ask astudenttoread the a class. share theirresponses as partner. Thenhavestudents table inQuestion1witha Have studentscompletethe Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 920 920

Problem

ExponentialandLogarithmicEquations Chapter 13 1. Recall thatalogarithmicfunctionistheinverseofanexponential. exponent inthecorresponding exponentialexpression . When youevaluatealogarithmicexpression (logarithm),youare determiningthevalueof and thenconvertitto logarithmicform. To writealogarithmicequation,sometimesitishelpfultoconsider theexponentialformfirst logarithmicequation .Alogarithmic equationisanequationthatcontainsalogarithm It isimportanttobecomefamiliarwithhowthebase,argument, andexponentfitintoa to 1;theargument mustbegreater than0;andthevalueofexponenthasnorestrictions . variables oftheexponentialequation.Thebase,b,mustbegreater than0butnotequal The variablesofthelogarithmicequationhavesamerestrictions as thecorresponding 1 Write theequivalentformofgivenexponentialorlogarithmicexpression .

It’s AllComingTogether . Exponential Form 10 e 12 3 5 (

<20.086 5100,000 __ 16 9 9 2 3 6 5

y 5 b 2 2 base __

3 2 2

x 5243 ) 536 8 3 5144 527 __ 2 1 5x 5 54 ___ 27 exponent x 8

5argument ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ log base (argument) 5exponent Logarithmic Form log100,00055 log log In 20.086<3 log log log log x 5 log 12 n __ 3 2 9 6 (243)5 16 (144)52 (27)5 2 (36)5x ( (x)58 log (4)5 ___ 27 8 ) b 53 (y) __ 2 1 __ 2 3

30/11/13 10:14AM 17/12/13 12:16PM © Carnegie Learning

© Carnegie Learning • • Question 3 for Share Phase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents parts ofQuestion3witha Have studentscompleteall grouping • • • • Question 2 for SharePhase, guiding Questions • • grouping solve fortheargument? What strategywasusedto argument? Why? base, theexponent,or Is iteasiertosolveforthe the argument? number represented How didyoudecidewhich the base? number represented How didyoudecidewhich the exponent? number represented How didyoudecidewhich logarithmic equation? before creating a exponential relationship Did youconsiderthe logarithmic equation. solve foranyunknownina demonstrating howto the workedexample As aclass,discuss a class. share theirresponses as partner. Thenhavestudents parts ofQuestion2witha Have studentscompleteall 451450_IM3_Ch13_917-988.indd 921

© Carnegie Learning • • • • If thebasesare equal,are theexponentsalwaysequal? If theexponentsare equal,are thebasesalwaysequal? What strategywasusedtosolveforthebase? What strategywasusedtosolvefortheexponent? exponential equation. To solveforanyunknowninasimplelogarithmicequation,beginbyconvertingittoan 2. 3. is unknown. equation toanexponentialregardless ofwhichterm It isimportanttonotethatyoucanconvertalogarithmic relationship tosolveforanyunknowninalogarithmicequation. Let’s considertherelationship betweenthebase,argument, andexponent. You can use that ruetI nnw Exponent Is Unknown Argument Is Unknown If c. 49,2,7 a. Arrange thegiventermstocreate atruelogarithmicequation. If 4 b. 4,1 c. If 4 a. Justify thelaststepofeachcaseinworkedexample. lo g 4 Intheequation4 thattheexponentsmustalsobeequal. calculatethevalueof4 Intheequation4 log log alsobeequal. Intheequation b 4 x 3 3 4 7 (y)53 64 5y 5 4 5y,whydoes64? 5 4 (4)51 (49)52 4 4 4 3 5y 1 7 7 54 2 3 3 549 , whydoesx53? , whydoesb54? b x 3 3 4 54 5y,thevariableisisolatedsoIcan 54 3 3 ,thebasesare ,theexponentsare 3 todeterminemyanswer. Exponential and Logarithmic Forms 13.1 lo g 4 (64)5x 256,4,4 d. b. 4 4 x 53 x x log 23, 6, log 5 4 564 equalsoIknow Exponential and Logarithmic Forms ExponentialandLogarithmicForms 13.1 4 6 (256)54 3 ( equalsoIknowthatthebasesmust

____ 216 1 6 6 ____ 216 4 4 23 1 4 )

5256 523 5

256 ____ 216 1 3 change as the unknown quantity differed? Base Is Unknown lo g may be helpful to think the strategy of a true exponential b How did (64)53 equation first! equation Remember, it b b b 54 3

564 5 4 3 first!

30/11/13 10:15AM 921

17/12/13 12:16PM 921 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 922 922

• • • • • • • • • • • • • • • Question 5 for Share Phase, guiding Questions • • Question 4 for SharePhase, guiding Questions share theirresponses asaclass. a partner. Thenhavestudents parts ofQuestions4and5with Have studentscompleteall grouping

lo g Is lo g What isthevalueoflo g Is lo g What isthevalueoflo g of lo g What istheexponentialform value oflo g Did youfirstdeterminethe What isthevalueoflo g Is lo g What isthevalueoflo g Is lo g What isthevalueofl Is lo g What isthevalueoflo g of lo g What istheexponentialform value oflo g Did youfirstdeterminethe equation inexponentialform? How canyourewrite the What isunknown? lo g lo g lo g How doyouknow? Exponential and Logarithmic Equations Chapter 13 5 5 5 7 (625)?Howdoyouknow? (625)?Howdoyouknow? (625)?Howdoyouknow?

(

2 __ 7 1 4 3 2 3 7 5

(256) equivalentto (81) equivalentto (16) equivalentto

( (625)?

) ( __

2 1 ( ? Howdoyouknow?

3 1 __ 1 7

)

equivalenttolo g

)

) equivalentto ? 7 5 (625)? (

__ 1 7

) ? og 4 5 (256)? (625)? 3 2 2 7 (81)? (16)?

7 ( (

__ __ 2 1 7 1 ( 7 __

) )

? ? )

? 13 451450_IM3_Ch13_917-988.indd 922 922 • • • •

What isthevalueoflo g What istheexponentialformoflo g Did youfirstdeterminethevalueoflo g What isthevalueoflo g

ExponentialandLogarithmicEquations Chapter 13 5. 4. log c. log First,Ievaluatedthegivenlogarithm. lo g a. Explain yourstrategy . Write three logarithmicexpressions thatare equivalenttoeachgivenexpression . lo g e. lo g a. Solve fortheunknownineachlogarithmicequation. log Sampleresponsesinclude: Answerswillvary. 8 n n ( ( ( 8 8 n n 2 logarithmicform.(Example:2 ThenIappliedtheexponentof4tothreeotherbases,andwrotethemin __ __ __ n52 2 1 2 1 2 1 n57 n n __ __ __ __ 2 2 2 2 3 3 3 3 8 58 564 5 5 5 49 5 n526 ) ) ) n n n n __ 8 5 1 2 5 2 5 52 564 ( (64)5n (625)

(64)5n (625)5x (16),log   7 ( 3 3 7

2 __ 2 3 ( 49 49  2  __ 2 1 __ 5 1 3 6 ) x54 __ 1 3 x ) 26 ) 5 5625

64 3

(

3 __ ( 3 1 __ 3 2 (81),log

) ? ) ? 4 (256),log 64

(

8 4 516,solog 64 ) ? __ 1 5 8? ( ____ 625 1 ) ,log10,000,ln( l og b. f logn523 d. lo g . n 3 3 n n 0.0015n 2 _____ 1000 9 9 (16)isequivalenttolog 2n 2 2 2 10 n5 n54 n 1 527 3 53 9 n 54 5 5 (27)5n 23 ( ___ 16 5n 5n 1 __ 3 2 __ ___ 4 16 1 1

3 2 2

) 52 e 4 ) 5 (625).) 30/11/13 10:15AM 17/12/13 12:16PM © Carnegie Learning

© Carnegie Learning lo g c. lo g b. lo g d. Thereisnovalueofxthatmakesthisequationtrue. log Sampleresponsesinclude: Answerswillvary. logarithmicform.(Example:(4) ThenIappliedtheexponentof 2 Thislogarithmisnotpossible.Theargumentmustbepositive. lo First,Ievaluatedthegivenlogarithm. log Sampleresponsesinclude: Answerswillvary. log First,Ievaluatedthegivenlogarithm. logarithmicform. ThenIappliedtheexponentof21tothreeotherbases,andwrotethemin x g 52 7 2 64 64 4 7 2

(22) (2),log ( (8) ( ( 64 (8)5x

__ 7 1 __ __ 2 1 7 1 7 7 x521

x5 x ) ) ) x

5x ,log 5 58 __ 7 1 __ 2 1 9 3 (3),log ( __ 3 1 ) ,log ( 3 Example:3 16 __ 4 1 (4),log0.1,ln (4),log Exponential and Logarithmic Forms 13.1 25 (5),log 21 __ 2 1 5 __ 2 1 52,solog tothreeotherbases,andwrotethemin __ 3 1 ( ,solog __ e 1 ___ 16 1 ) ( Exponential and Logarithmic Forms ExponentialandLogarithmicForms 13.1 __ 1 4 ) ,ln( 3 4 (2)isequivalenttolog ( __ 1 3 e ) __ isequivalenttolog 1 2 ) 64 7 (8).) ( __ 1 7 ) . )

30/11/13 10:15AM 923

17/12/13 12:16PM 923 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 924 924

base oflogarithms. number linetoestimatethe the exponent,studentsusea Next, giventheargument and • • • • • • Questions 1and2 for SharePhase, guiding Questions • • grouping that lo g provided. Studentsgeneralize base. Aworkedexampleis argument isnotapowerofthe to thetenthsplacewhen estimate thevalueoflogarithms Students useanumberlineto Problem 2

argument of1? equivalent to0allhavean Do thelogarithmsthatare equivalent to0? arguments ofalllogarithms What istrueaboutthe the logarithmsof base 5? labels onthenumberlinefor How didyoudeterminethe the logarithmsof base 4? labels onthenumberlinefor How didyoudeterminethe the logarithmsofbase3? labels onthenumberlinefor How didyoudeterminethe the logarithmsofbase2? labels onthenumberlinefor How didyoudeterminethe as aclass. Discuss theworkedexample a class. share theirresponses as partner. Thenhavestudents Questions 1and2witha Have studentscomplete Exponential and Logarithmic Equations Chapter 13 b

(1) 5 0andlo g b

( b ) 5 1. 1. 13 451450_IM3_Ch13_917-988.indd 924 924 • • • •

How is How is to the bases? Do thelogarithmsthatare equivalentto1allhavearguments thatare equal What istrueaboutthearguments ofalllogarithmsequivalent to1? Problem

ExponentialandLogarithmicEquations Chapter 13 b b 1 0 5bwrittenasalogarithmicequation? 51writtenasalogarithmicequation? 1. number, evenanirrationalnumber . In Problem 1,eachlogarithmwasarationalnumber .However, alogarithmcanbeany real 2. 2 Label eachnumberlineusinglogarithmicexpressions withtheindicatedbase. Analyzeallthelogarithmsthatare equivalentto0. a. Compare thelogarithmsonnumberlines. Analyzeallthelogarithmsthatare equivalentto1. b. Logarithms of Logarithms of Logarithms of Logarithms of

base 4 base 3 base 2 base 5 log argumentof1,regardlessofthebase.Therefore, Thelogarithmsthatare this relationship . Write ageneralstatementusingthebasebtorepresent log argumentthatisequaltothebase.Therefore, Thelogarithmsthatare represent thisrelationship . Write ageneralstatementusingthebasebto Estimating Logarithms b b (b)51for.0,ﬁ1. (1)50forb.0,ﬁ1. log log log log 2 4 5 3 2 _ + _ _ + _ 25 16 _ + _ _ + _ 1 1 9 1 4 1 log log log log 21 4 5 3 equivalentto0allhavean equivalentto1allhavean _ + _ _ + _ 2 _ + _ 4 1 5 1 _ + _ 3 1 2 1 log log log log 0 3 4 2 5 (1) (1) log (1) (1) log log log 1 4 3 2 5 (4) log (3) (2) (5) log 5 (18) log log log 5 4 2 2 3 (25) (16) (4) ()log (9) log log 4 (18) 3 (18) log log log 5 4 3 (125) 3 2 when appropriate. (4 log (64) (27) ()log (8) restrictions on the variables Describe the log log log 5 4 3 2 (625) (256) 2 4 (81) (16) (18) 30/11/13 10:15AM 17/12/13 12:16PM © Carnegie Learning

© Carnegie Learning as aguide. You canestimatethevalueofalogarithmthatisnotanintegerbyusingnumberline Rewritethegeneralstatementsfrom parts(a)and(b)inexponentialform. c. The Use exponentrulestoverifythateachstatementistrue. b statement Any The is closerto3than4. To estimatelo g Estimate thevalueoflo g In thiscase,3.2isagoodestimateforlo g than 33onthenumberlinethatrepresents base3. argument islessthan33andtheclosestlogarithmwhoseargument isgreater is between3and4. You knowthatlo g Because 33iscloserto27than81,thevalueoflo g Next, estimatethedecimaldigit. 1 5 The closestlogarithm whose argument is logarithmic logarithmic base b . less than33: Any lo g raised is base log 3 true. (27) 3 3 (27) 3 equation equation (33)tothetenthsplace,identifyclosestlogarithmwhose to raised log 3 2753andlo g the0 3 (33) to 3 power lo g log log (33). Exponential and Logarithmic Forms 13.1 the 3 ,x 3 b b (27),lo g The logarithmyouare ( (1) 1st b is ) 5 5 equal estimating: 3 power 0 8154.Thismeanstheestimateoflo g 1 lo g can can 3 3

Exponential and Logarithmic Forms ExponentialandLogarithmicForms 13.1 (33) (33),lo g to be is 3 be (33). 1, log equal rewritten , rewritten so 3 4 (81) 4 the to 3 (81) logarithmic itself, 3 in 33 in The closestlogarithm exponential exponential whose argument is so greater than33: 3.1 and 3.2 are both good is just an estimate, so the lo g Remember, this statement logarithmic 3 (81) answers. form form as 3 as 33 is b true. 0 5 1.

30/11/13 10:15AM 925

17/12/13 12:16PM 925 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 926 926

• • • Question 3 for SharePhase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents Questions 3and4witha Have studentscomplete grouping

making theirestimation? definition oflogarithmwhen Did SuttonorSilasusethe lo g The valueofthelogarithm number line? two logarithms onthe located betweenwhich The logarithmlo g the number line? two logarithmic valueson Exponential and Logarithmic Equations Chapter 13 4 (28)isbetweenwhich 4 (28)is 13 451450_IM3_Ch13_917-988.indd 926 926

ExponentialandLogarithmicEquations Chapter 13 3

Sutton andSilaswere eachaskedtoestimatelo g logarithmwhenmakinghisestimate. ThemethodSilasusedwillalwaysworkbecauseheisusingthedefinitionofa Will Silas’s methodalwayswork? Silas didnotusethenumberline,buthisestimatewasaboutsameasSutton’s . I knowthat 4 I estimatedlo g I estimated lo g between 2and3. and estimatingbasedonpowersof4. Silas Sutton log 2 4 1 16 1

2 2 4

log 4 5 (28) by using the number line. 21 (28)byconvertingthelogintoexponential form 16and 4 4 1 4 1 lo g 2 4 (16) log 2 0 4

3 ()log (1)

lo g lo g , , 5 lo g x lo g 4 64sotheestimateoflo g 4 (28) (28)<2.4 4 (28)5 4 (28) 1 4 4 ()log (4) x ¯ 528 2.3 , , x 3 lo g 4 2 (6 log (16) 4 (64) 4 (28). 4 3 (4 log (64) 4 (28)mustbe 4 (256) 4 30/11/13 10:15AM 17/12/13 12:16PM © Carnegie Learning

© Carnegie Learning • • • • • • • • Question 4 for Share Phase, guiding Questions number line? two logarithmicvaluesonthe log 2500isbetweenwhich The valueofthelogarithm number line? two logarithmsonthe located betweenwhich The logarithmlog2500is lo g The valueofthelogarithm number line? two logarithms onthe is locatedbetweenwhich The logarithmlo g lo g The valueofthelogarithm number line? two logarithms onthe located betweenwhich Thelogarithmlo g lo g The valueofthelogarithm number line? two logarithms onthe located betweenwhich The logarithmlo g number line? two logarithmicvaluesonthe the number line? two logarithmic valueson the number line? two logarithmic valueson 4 5 2 (300) isbetweenwhich (4)isbetweenwhich (10)isbetweenwhich 4 2 5 (300) (10)is (4)is 451450_IM3_Ch13_917-988.indd 927

© Carnegie Learning 4 . lo g a. explain yourreasoning . Estimate eachlogarithmtothetenthsplaceand lo g c. log2500 d. lo g b. log log log Thelogarithmlog Answerswillvary. Thelogarithmlog Answerswillvary. Thelogarithmlog Answerswillvary. to3thanto4. Since2500iscloserto1000thanto10,000,theapproximationshouldbecloser Since4iscloserto5thanto1,theapproximationshouldbecloserto1thanto0. log1000,log2500log10,000 Thelogarithmlog2500isapproximatelyequalto3.4. Answerswillvary. 4thanto5. Since256iscloserto300thanto1024,theapproximationshouldbecloserto shouldbecloserto3thanto4. Since10iscloserto8thanto16,theapproximation 0 , x 4 2 4 5 3 5 2 4 3 (300) (4) (10) (1),log (8),log (256),log ,x ,4 ,x ,x,5 5 2 (4),log (10),log 4 (300),log ,1 2 4 5 300isapproximatelyequalto4.1. 4isapproximatelyequalto0.8. 10isapproximatelyequalto3.3. ,4 5 (5) 2 (16) Exponential and Logarithmic Forms 13.1 4 (1024) Exponential and Logarithmic Forms ExponentialandLogarithmicForms 13.1 lines from Question 1 to help you! use the number You can

30/11/13 10:15AM 927

17/12/13 12:16PM 927 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 928 928

• • • • • • Question 5 for Share Phase, guiding Questions share theirresponses asaclass. a partner. Thenhavestudents Questions 5through 11with Have studentscomplete grouping

of 125? 58 close totheupperlimit When thebaseis5, of 64? 58 closetotheupperlimit When thebaseis4, upper limit? 58 should beclosertoits that meantheargument exponent is2.9,does Because thevalueof exponent inthissituation? What isthevalueof this situation? correct answerin Is there more thanone be correct? Can MarkandScottyboth Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 928 928

ExponentialandLogarithmicEquations Chapter 13 ? 5 . Whenthebaseis4,58isverycloseto4 shouldbeclosetoitsvaluewhen3istheexponent. Becausethevalueoftheexponentis2.9,thatmeansthattheargument Markiscorrect. Who iscorrect? Explainyourreasoning . estimate thevalueoflo g Mark andScottywere eachaskedtodeterminewhichbasewasused iscorrect. Theestimateof2.9makesmoresensewhenthebaseis4.Therefore,Mark 58isnotverycloseto5 So, b lo g when the base is 4. The log of 58 falls between 2 and 3 Mark log log 4 2 2 (16) 2 4 5 ¯ 1 1 16 25 1 1

≈ 4. 2 2 , , 2.9 lo g b (58) log log 21 21 4 5 1 1 4 1 5 1 2 2 , , 3 lo g b 3 4 ,or125. (58)52.9. (64) log log 0 0 4 5 (1) (1) lo g when thebaseis5. logof58fallsbetween2 The Scotty So, b So, 5 (25) log log 2

3 1 1 4 5 ,or64,whereaswhenthebaseis5, (4) (5) ¯ 5. , , lo g

log log

4 5 2 2 (6 log (16) (5 log (25) b 2.9 (58) log 5 (58) log , , 4 5 (58) 4 (125) 3 3 lo g

(64) 3 5 (125) and 3 log log 5 4 (625) (256) 4 4 30/11/13 10:15AM 17/12/13 12:16PM © Carnegie Learning

© Carnegie Learning • • • • • • • • • Question 6 for Share Phase, guiding Questions in thissituation? of 5beappropriate associated withthebase won’t thenumberline If theargument is74,why if theexponentis3.1? associated withthelogof74 What basenumberis number line? of 74fallbetweenonthe two valuesdoesthelog If theargument is74,what this situation? of 3beappropriate in associated withthebase why won’t thenumberline If theargument is0.4, 0.4 iftheexponentis21.3? associated withthelogof What basenumberis number line? of 0.4fallbetweenonthe two valuesdoesthelog If theargument is0.4,what this situation? of 4beappropriate in associated withthebase why won’t thenumberline If theargument is108, 108 iftheexponentis2.9? associated withthelogof What basenumberis number line? of 108fallbetweenonthe two valuesdoesthelog If theargument is108,what 451450_IM3_Ch13_917-988.indd 929

© Carnegie Learning 6 . lo g b. lo g a. each logarithm. Use thenumberlinesfrom Question1todeterminetheappropriate baseof lo g c. log b¯2 log b¯5 log b¯4 exponentis3.Therefore,log Sincethevalueis3.1,theargument74shouldbeclosetothevaluewhenthe log and log log log 2 2 2 2 b b b log 2 5 4 3 4 3 (0.4)521.3 (108)52.9 (74)53.1 (0.25),log (25),log (64),log (27),log 5 22, 3, 3, 2,x 1 1 16 9 1 1 1 2 25

2 1

1 2 4 1 2 2 log 2 (0.4) log log log log 5 4 3 x x 21 21 21 21 (108),log (74),log (74),log 3 2 4 2 x 5 (0.4),log 1 1 1 3 1 1 4 1 2 1 2 5 1 2 2 2 ,4 ,4 ,3 ,21 4 3 Exponential and Logarithmic Forms 13.1 log log log (256) (81) log 5 (125) 2 0 0 0 0 2 4 3 (0.5) 5 3 (1) (1) (1) (1) (74)doesnotmakesense.Thebasemustbe4. log log log log Exponential and Logarithmic Forms ExponentialandLogarithmicForms 13.1 1 1 1 1 2 4 3 5 (2) (4) (3) (5) log log log log 4 5 2 2 2 2 2 3 (6 log (16) (5 log (25) ()log (4) ()log (9) log 5 (108) 5 log 4 3 3 3 3 3 2 (2)log (125) (64) (27) ()log (8) 4 (74) log 3 log log (74) 4 5 2 3 (256) 4 4 4 4 (625) (16) (81)

30/11/13 10:15AM 929

17/12/13 12:16PM 929 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 930 930

• • • • • • • • • • • • • Questions 7through11 for SharePhase, g uiding Questions uiding Questions

Is 2.718closerto2or3? 4.2 butgreater than 2.6? Is thevalueofln18less than What isthevalueoflo g What isthevalueoflo g 2.718 between? Which twologvaluesis exponential form? What isln18writtenin of ln18? What isthevalueofbase or smaller? the logarithmgetlarger larger, doesthevalueof the valueofbasegets For afixedargument, as is lo g Which twonumericvalues is lo g Which twonumericvalues is lo g Which twonumericvalues is lo g Which twonumericvalues or smaller? the logarithmgetlarger larger, doesthevalueof value oftheargument gets For afixedbase,asthe the tenthsplace? determine theestimateto number line?Howdidyou the tenthsplace? determine theestimateto number line?Howdidyou the tenths place? determine theestimateto number line?Howdidyou the tenths place? determine theestimateto number line?Howdidyou Exponential and Logarithmic Equations Chapter 13 5 4 3 2 (18) betweenonthe (18) betweenonthe (18)betweenonthe (18)betweenonthe

3 2 (18)? (18)?

13 451450_IM3_Ch13_917-988.indd 930 930

ExponentialandLogarithmicEquations Chapter 13 7 11. 8 10 1 9 . . . . SinceIknowthatlog (2.718) Forexample,log getslargeraswell. Forafixedbase,asthevalueoftheargumentgetslarger,thevalueofthelogarithm to thevalueoflogarithm?Provide anexampletoillustrateyourstatement. For afixedbasegreater than1,asthevalueofargument getslarger, whathappens In18¯2.9 Since2.718iscloserto3thanto2,thevalueofthelogarithmmustbecloserto2.6. Answerswillvary. Make aprediction forthevalueofIn18. logarithmwillhavetobelessthan4.2butgreaterthan2.6. Plot lo g astheargumentincreases,sodoesthevalueofthelogarithm. log Seenumberlines. to thetenthsplace. Verify youranswersinexponentialform. Question 1.Thenusethenumberlinestoestimatenumericvalueofeachlogarithm How couldyouusethenumberlinesfrom Question1topredict thevalueofIn18? getssmaller. Forafixedargument,asthevalueofthebasegetslarger,thevalueofthelogarithm happens tothevalueoflogarithm? For afixedargument, whenthevalueofbaseisgreater than1andincreasing, what log log log ln185x 5 4 3 2 (18)¯1.8 (18)¯2.1 (18)¯2.6 (18)¯4.2 e x x 518 ¯18 2 (18),lo g 3 3 (18),lo g (3)51,log 2 (18)¯4.2,andlog 4 (18),andlo g 3 (9)52,log 5 4 3 2 Inexponentialform,thismeansthat 5 (18)ontheappropriate numberlinesin 1.8 2.1 2.6 4.2 3 3 (18)¯2.6,Ialsoknowthatthevalueofthe ¯18 ¯18 ¯18 ¯18 (27)53.Thebasestayedfixedat3,but 30/11/13 10:15AM 17/12/13 12:16PM © Carnegie Learning

© Carnegie Learning • • • • • • talk thetalk for Share Phase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents Questions 1through 8with a Have studentscomplete grouping never’ tomaketruestatements. words ‘always,sometimes, or Students choosefrom the talk thetalk logarithm equal1? Why can’t thebaseofa be equaltoanyreal number? Can thevalueofalogarithm of alogarithm? associated withthebase What restrictions are of logarithms? What istherange of logarithms? What isthedomain of logarithm? What isthedefinition 451450_IM3_Ch13_917-988.indd 931

© Carnegie Learning Talk theTalk 3. 2. 4. 5. 6. 7. 8. Be prepared toshare yoursolutionsandmethods. Choose aword from theboxthatmakeseachstatementtrue.Explainyour reasoning . 1. The valueofalogarithmis cannotbenegative. Abasiclogarithmicfunctionhasadomainof(0,`).Theargumentstherefore The argument ofalogarithmicexpression is Thisisthedefinitionofalogarithm. exponential equation. The baseofalogarithmis Therangeofalogarithmis(2`,`). A logarithmis notequalto1. Bydefinition,thebaseofalogarithmicexpressionisgreaterthanzeroand log For abasegreater than1,ifb.cthenthevalueoflo g realnumberfrom(2`,`). Becausethevalueofalogarithmistheexponentonthebase,itcanequalany If a.b,thenthevalueoflo g thelogarithm. Whenthebasesare The baseofalogarithmis Thisisbasicallyaconstantfunctionbecause1toanypowerisequalto1. than lo g equalto0.Thuslog Anylogarithmicexpressionwithanargumentof1,regardlessofthebase,willbe The valueofalogarithmis 1 (y)5x→1 a c. always sometimes a thesame,astheargumentgetslarger,sodoesthevalueof 1,willneverbelessthanlog x 5y a sometimes Exponential and Logarithmic Forms 13.1 1is avaluethatisnotwholenumber . always never never sometimes never anegativenumber . equalto1. equaltotheexponentofcorresponding equaltoanegativenumber . Exponential and Logarithmic Forms ExponentialandLogarithmicForms 13.1 never lessthanlo g b 1,becausetheyare a bis never anegativenumber . b 1. always equal. greater

30/11/13 10:15AM 931

17/12/13 12:16PM 931 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 932 932

Check forStudents’Understanding

Exponential and Logarithmic Equations Chapter 13 place. Verify youranswersinexponential form. log Use numberlinestoestimatethenumericvalueoflo g 2 2 log log log 5 5 4 3 1.4 1.7 2.1 3.3 ¯10 ¯10 ¯10 ¯10 5 4 3 2 (10)¯1.4 (10)¯1.7 (10)¯2.1 (10)¯3.3 0 1 2 2 (10),lo g 3 3 log (10),lo g 2 (10) 4 (10),andlo g 4 5 (10)tothetenths 17/12/13 12:16PM

© Carnegie Learning E In thislesson,youwill: lEArning gOAlS ofl Properties m • • • • • • • SSEntiA base ofthepower, orlo g of theexponentandlogarithm logarithm ofapowerisequaltotheproduct The PowerRuleofLogarithmsstates:“The and thedivisor, orlo g difference ofthelogarithmsdividend “The logarithmofaquotientisequaltothe The QuotientRuleofLogarithmsstates: lo g sum ofthelogarithmsfactors,or “The logarithmofaproduct isequaltothe The Product RuleofLogarithmsstates: always equaltozero, orlo g “The logarithmof1,withanybase,is The Zero Property ofLogarithmsstates: a singlelogarithmicexpression. Rewrite multiplelogarithmicexpressions as properties oflogarithms. Expand logarithmicexpressions usingthe Derive theproperties oflogarithms. ad Props b (xy)5lo g l id b (x)1lo g EAS b

(

y __

) b

5 (y).” b ( lo g x b n 150.” )5nlo g b ogarithms

( x ) 2 lo g b (x).” b

( y ).” St CO KEy tErmS Build newfunctionsfrom existingfunctions F-BF BuildingFunctions • • • • • 5. A Power RuleofLogarithms Quotient RuleofLogarithms Product RuleofLogarithms Logarithm withSameBaseandArgument Zero Property ofLogarithms mmO involving logarithmsandexponents. use thisrelationship tosolveproblems between exponentsandlogarithms (+) Understand theinverserelationship ndA n COrE rdS fO r mAthEmA St AtE 13.2 tiCS 933A 17/12/13 12:16PM 13 451451_IM3_TIG_Ch13_917-988.indd 2 933B

Exponential and Logarithmic Equations Chapter 13 exponential andlogarithmicproperty verballyandsymbolically, providing examplesforeachinstance. rules andproperties. They summarizethedifferent properties bycompletingatablewhichdefineseach Students developrulesandproperties oflogarithmsbasedontheirpriorknowledgevariousexponent Overview 17/12/13 12:16PM

TheProductRuleofExponents

___ ( ( ( Anynumberraisedtothefirstpowerisequaltoitsbasenumber. TheQuotientRuleofExponents 2 2

______2 2 3 3 2 3 14 5

) ) )

0 1 5

5 2

51 5 ? (

__ 2 3 (

9 __ 3 2

)

) 5

(

__ 3 2

) 7

Properties of Logarithms 13.2

933C 17/12/13 12:16PM 13 13 451451_IM3_TIG_Ch13_917-988.indd 4 933d

Exponential and Logarithmic Equations Chapter 13 17/12/13 12:16PM

© Carnegie Learning created a table that probably embellishes every chemistry classroomintheworld. chemistry created atablethatprobablyembellishesevery ofatomsand Dmitri Mendeleevrevolutionizedourunderstandingoftheproperties not includeanyofthenoblegases,however, whichhadnotyetbeendiscovered. elements andpointedoutacceptedatomicweightsthatwereinerror. Histabledid ofnew Hepredictedtheexistenceandproperties them bysimilarityofproperties. theelementsinascendingorderofatomicweightandgrouping the basisofarranging in1869.Hisfirstperiodictablewascompiledon published inPrinciplesofChemistry In thislesson,youwill: LEaRning goaLS ofLogarithms Properties Mad Props D • • • a singlelogarithmicexpression. Rewrite multiplelogarithmicexpressions as properties oflogarithms. Expand logarithmicexpressions usingthe Derive theproperties oflogarithms. 63 knownelementsintoaperiodictablebasedonatomicmass,whichhe the imitri Mendeleevisbestknownforhisworkontheperiodictable—arranging KEY TERMS • • • • • Power RuleofLogarithms Quotient RuleofLogarithms Product RuleofLogarithms Logarithm withSameBaseandArgument Zero Property ofLogarithms Properties of Logarithms 13.2 13.2 05/12/13 4:41PM 933

17/12/13 12:16PM 933 13 13 451451_IM3_TIG_Ch13_917-988.indd 934 934

• • • • Questions 1and2 for SharePhase, guiding Questions share theirresponses asaclass. a partner. Thenhavestudents parts ofQuestions1and2with Have studentscompleteall grouping logarithmic property lo g Students alsoderivethe the Zero Property ofPowers. Property ofLogarithmsusing Students developtheZero Problem 1 of Logarithms. to developtheQuotientRule Quotient RuleofPowersisused Rule ofLogarithms,andthe used todeveloptheProduct The Product RuleofPowers is property using thecorresponding

are equal,alwaysequalto1? when thebaseandargument Is thelogarithmofanumber the base? the firstpowerequalto Is anynumberraisedto base, alwaysequalto0? Is thelogarithmof1,withany power equalto1? Is anybaseraisedtothezero Exponential and Logarithmic Equations Chapter 13 b 1 5bforexponents. b (b)51 13 451450_IM3_Ch13_917-988.indd 934 934

Problem

ExponentialandLogarithmicEquations Chapter 13 1. correspond tovariousexponentialrulesandproperties youalready know . exponents andpowers.Inthislesson,youwilldeveloplogarithmicrulesproperties that Logarithms bydefinitionare exponents,sotheyhaveproperties thatare similartothoseof 2. 1 Write asentencetosummarizetheZero Property a. corresponding logarithmicproperty . Let’s considertheZero Property ofPowerstodevelopa Write theZero Property ofPowersinlogarithmic b. Write yourexponentialequation from part(a)inlogarithmicform. b. Write anexponentialequationtorepresent thisrule.Usebasthebase a. equal tothebase. Let’s considertheexponentrulethatsaysanynumberraisedtofirstpoweris StatetheZero Property ofLogarithmsinwords . c. Statethislogarithmicrelationship inwords . c.

of Powers, log property calledtheZeroPropertyofLogarithms. form .Thisisacorresponding logarithmic Anybaseraisedtothezeropoweris1. log b Thelogarithmof1,withanybase,isalwaysequalto0. Whenthebaseandargumentare Setting groundrules 1 5b b b (1)50 (b)51 b 0 51. equal,thelogarithmisalwaysequal to1. use your number lines to verify representations. How did you at your number line this property? Look back 30/11/13 10:15AM 17/12/13 12:16PM © Carnegie Learning

© Carnegie Learning • • Question 3 for SharePhase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents parts ofQuestion3witha Have studentscompleteall grouping of thefactors? difference ofthelogarithms product equaltothesum or Is thelogarithmofa divide theexponents? add, subtract,multiply, or with thesamebase,doyou When multiplyingpowers 451450_IM3_Ch13_917-988.indd 935

© Carnegie Learning equation . as alogarithmic as alogarithmic equation exponential Rewrite each 3. AnalyzethestepsthatbeginwithProduct Ruleof b. StatetheProduct RuleofLogarithmsinwords . c. Write asentencetosummarizetheProduct RuleofPowers, a. logarithmic property . Let’s considertheProduct RuleofPowerstoderiveacorresponding

Given: Given: Complete thelastlineofdiagram. property calledtheProductRuleofLogarithms. Powers toderiveacorresponding logarithmic Whenmultiplyingpowerswiththesamebase,youaddtheexponents. Thelogarithmofaproductisequaltothesumofthelogarithmsofthefactors. log Let x5b

b x (x) 5m ( x 5 ) b b

5 b

⇔ m m ? m ? . Finally, usesubstitution terms ofxandy: to writetheproperty in Finally, usesubstitution bb

n 5 5 bb m1n Lety5b

log 1 n b y (y)5n ( y 5 ) 5

⇔ n .. n lo g Properties of Logarithms 13.2 Use substitutionto rewrite thegiven: b (xy)5 Properties of Logarithms Properties 13.2 log b (x)1log lo g lo substitution to make the property easier to derive. g b b m (xy)5m1n ? xy

xy 5 start with a b ) n You 5 5 b ( can y) b m b m1n ⇔ m1n 1 . n

30/11/13 10:15AM 935

17/12/13 12:16PM 935 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 936 936

• • • • • Questions 4and5 for SharePhase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents Questions 4and5witha Have studentscomplete grouping

of thebasepower? exponent andthelogarithm equal totheproduct ofthe Is thelogarithmofapower added together or multiplied? a power, are theexponents When simplifyingapowerto change orremain thesame? to apower, doesthebase When simplifyingapower and thedivisor? logarithms ofthedividend or thedifference ofthe logarithms ofthefactors equal tothesumof Is thelogarithmofaquotient the exponents? subtract, multiply, ordivide the samebase,doyouadd, When dividingpowerswith Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 936 936

equation . as alogarithmic equation exponential Rewrite each ExponentialandLogarithmicEquations Chapter 13 4. CompletethestepsthatbeginwithQuotient b. Write asentencetosummarizetheQuotientRuleofPowers, a. logarithmic property . Let’s considertheQuotientRuleofPowerstoderiveacorresponding SummarizetheQuotientRuleof Logarithmsinwords . c.

Given: Given: ofLogarithms. logarithmic property calledtheQuotientRule Rule ofPowerstoderiveacorresponding Whendividingpowerswiththesamebase,yousubtracttheexponents. log dividendandthedivisor. Thelogarithmofaquotientisequaltothedifferenceofthelogarithmsofthe Let x5b

b x x 5m 5 ___ b

bb b

⇔ m n m 5

5 .. terms ofxandy: to writetheproperty in Finally, usesubstitution b b m2n n , ifnﬁ0 Lety5b

log n b ﬁ y y5n 0 5

⇔ n . lo g

b rewrite thegiven: Use substitutionto (

__ y x ) 5 log b x2log Product Rule of Logarithms to help you get started. your derivation of the ___ b b m n b 5

y log b Look at m2n b __ x y ( , if 5b __ x y ) 5m2n b n m2n

⇔ ﬁ0. 30/11/13 10:15AM 17/12/13 12:16PM © Carnegie Learning

StatethePowerRuleofLogarithms inwords . c. Write asentencetosummarizethePowerRule,( a. logarithmic property . Let’s considerthePowertoaRulederivecorresponding

CompletethestepsthatbeginwithPowertoaRulederive b. Let x5b

logarithmofthebaseofthepower. Thelogarithmofapowerisequaltotheproductoftheexponentandthe terms ofx: to writetheproperty in Finally, usesubstitution Given: Given: ( corresponding logarithmicproperty calledthePowerRuleofLogarithms. To simplifyapowertoapower,keepthebaseandmultiplytheexponents. x log 5 ( b b b

b x5m

⇔ m m ) .. n 5 5 b mn

Use substitutionto rewrite thegiven: lo g b x n 5

Properties of Logarithms 13.2 n?log Properties of Logarithms Properties 13.2 b x b m ) n log 5

x b n b 5b x mn

n ⇔ 5mn . mn

30/11/13 10:15AM 937

17/12/13 12:16PM 937 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 938 938

• • • • • • • • • • Question 6 for Share Phase, guiding Questions responses asaclass. Then havestudentsshare their Question 6withapartner. Have studentscomplete grouping

which property? of whichproperty? lo g ( lo g __ lo g x lo g 7 lo g 1 5 which property? which property? which property? which property? example ofwhichproperty? which property? property? is anexampleofwhich property? is anexampleofwhich x x x 2 3 9 1 3 ?

Exponential and Logarithmic Equations Chapter 13

5 57isanexampleof ) 0 6 6 6 6 6 6 51isanexampleof 5 (125) 53lo g (54) 2lo g (5) 1 lo g (6)51isanexampleof (1)50isanexampleof x 6 x ? 6 x isan exampleof x 18 5 isan exampleof 5 x 6 13 (4) 5 lo g 6 (6) 5 lo g isan example 6 (5) is an 6 (20) 6 (9) 13 451450_IM3_Ch13_917-988.indd 938 938

ExponentialandLogarithmicEquations Chapter 13 6. Answerswillvary. Provide examplesforeachproperty . to defineeachexponentialandlogarithmicproperty verballyandsymbolically . In thisproblem, youderiveddifferent properties oflogarithms.Completethetables Examples: Symbolic: Anybaseraisedtothezeropoweris1. Verbal: Examples: baseitself. Anybaseraisedtothefirstpoweristhe Verbal: Symbolic: youaddtheexponents. To Verbal: Symbolic: Examples: multiplypowerswiththesamebase, aeRie oFrtPwrLogarithm with Same Base and Argument Base Raised to First Power Zero Property of Powers Product Rule of Powers Exponential Property 3 z z 4 m ? ?3 b z m 7 ? n 5 10 1 m m ?3 p p b b 6 b 1 1 1 0 0 0 5b 56 5p n 51 z 51 51 5 p 11 53 b m1n m1np equalto0. Thelogarithmof1,withanybase,isalways Verbal: log Examples: Symbolic: Symbolic: The Verbal: to sumofthelogarithmsofthefactors. Thelogarithmofaproductisequaltothe Verbal: log Symbolic: Examples: Examples: the a 2 (50)5log (mnp)5log logarithm same Zero Property of Logarithms Product Rule of Logarithms log Logarithmic Property b number, (xy)5log ofa 2 (5)1 log a (m)1log log log log log log log number, d b b 2 7 a is (d)51 (b)51 (2)51 (1)50 (1)50 (1)50 always b (x)1log 2 (10) with a (n)1log equal the b ( base y) to a (p) 1. equal 30/11/13 10:15AM 17/12/13 12:16PM © Carnegie Learning

© Carnegie Learning • • Questions 1through3 for SharePhase, g share theirresponses asaclass. partner. Thenhavestudents Questions 1through 3witha Have studentscomplete grouping various logarithmicexpressions. algebraic expressions for logarithm. Studentswrite expressions asasingle form andtowritelogarithmic expressions inexpanded logarithms towritelogarithmic Students useproperties of Problem 2 uiding Questions uiding Questions difference oflo g lo g Does theexpandedformof lo g Does theexpandedformof 4 lo g 5 lo g difference oflo g 7 4 (6 (

7 6 ___ 3 (y),and3lo g (x)? x x y 3 5 4

) involve thesumor

) involvethesumor 7 4 (6)and (3), 7 (x)? 451450_IM3_Ch13_917-988.indd 939

© Carnegie Learning • • Problem and 3lny? a singlelogarithm? Which ruleorproperty wasusedtowritethelogarithmicexpression as Does theexpandedformofln(3x y 1. 2 In(3 c. lo g b. lo g a. expression inexpandedform. Use theproperties oflogarithmstorewrite eachlogarithmic Symbolic: yousubtracttheexponents. To Verbal: Examples: Symbolic: baseandmultiplytheexponents. To Verbal: Examples:

dividepowerswiththesamebase, simplifyapowertoapower,keepthe log In(3 log Don’t Breaktherules! 7 4 7 4

(6 (6 xy xy ( (

___ 3 ___ Quotient Rule of Powers 3 x x Power to a Power Rule x y x 3 y 3 3 Exponential Property 3 ) 5 ) 4 5

4 5In(3)1In(x )

)

) ___ ) b 5log

b 5log m n 5 (b 7 ( ( ___ 5 5 a b m __ r r r d w f 3 4 ) m2n ) 7 5 8 55 ) (6 4 n v 5 (3 5 57 5 , ) ) ifnﬁ0 15log r 14log b d2f b 3 wv 12 mn ) 13In( 3 ) involvethesumordifference ofln3,x, 4 7 (x ( y ) ) 23log y ) andthedivisor. differenceofthelogarithmsofthedividend Thelogarithmofaquotientisequaltothe Verbal: of thebaseofthepower. productoftheexponentandthelogarithm Thelogarithmofapowerisequaltothe Verbal: Examples: Symbolic: Symbolic: Examples: 7 Properties of Logarithms 13.2 ( x ) Quotient Rule of Logarithms log log Power Rule of Logarithms log Properties of Logarithms Properties 13.2 a Logarithmic Property 2 b lo log (5)5log ( __ m ( log n g __ x y b ) ) 2 5log ( 5log (27)53log a x ( n r )5n?log s )5slog 2 b a (45)2 log (x)2log (m)2log properties apply to common logarithms! logarithms and The logarithm both natural a 2 b (r) (3) (x) b 2 ( a (9) (n) y)

30/11/13 10:15AM 939

17/12/13 12:16PM 939 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 940 940

• • • Questions 2and3 for SharePhase, guiding Questions

or (2p1r)? or (q2rp)? ( Is lo g Is lo g Is lo g or 23q? q

Exponential and Logarithmic Equations Chapter 13 1

r a a a

(0.3) equal to (0.3)equalto (50)equalto(2p2r) 2 (

___ 27 1

) or (q2r1p) ) equalto3q 13 451450_IM3_Ch13_917-988.indd 940 940

ExponentialandLogarithmicEquations Chapter 13 2. 3. Be prepared toshare yoursolutionsandmethods. lo g a. single logarithm. Use theproperties oflogarithmstorewrite eachlogarithmicexpression asa 4log(12)2(2) b. 3(In32Inx)1(In9) c. Suppose lo g lo g b. lo g c. lo g a. each logarithmicexpression . log log 4log(12)24log(2 log 3(In32Inx)1(InIn9)5 log 2 a a a 2 a a a (50) (0.3) (10)13lo g (50)5log (10)13log (0.3)5log ( (

___ 27 ___ 27 1 1

) )

5log 52p1r 52log 523q 5log a 5log 5q2rp (5)5p,lo g a a a a a (5 (1)23log (3)2(log 2 ( a 2 ( (x) __ 3 (5)1log (x ___ 10 1 2 3 3 ?2) ) ) ) ) 5log 5log a (3)5q,andlo g a 2 a (2)1log a (10 (2) (3) ( ___ 12 2 4 4 x ) 3 5log(1296)¯3.1126 ) ( __ 3 x ) a 3 (5)) ( __ 9 x a (2)5r.Write analgebraicexpression for ) 5In ( __ x 3 2 ) 30/11/13 10:15AM 17/12/13 12:16PM © Carnegie Learning

log 2 2

______2 2 3 3 2 3 14 5

) ) )

0 1 5

5 2

a a a a 51 5 ? (1)50 (a)51 (xy)5log ( __ x y (

__ 2 3 ( )

9 __ 5log 3 2

)

) 5

(

__ 3 2 a

) (x)2log a 7 (x)1log

a ( a ( y) y) Properties of Logarithms 13.2

940A 17/12/13 12:16PM 13 13 451451_IM3_TIG_Ch13_917-988.indd 2 940B

Exponential and Logarithmic Equations Chapter 13 17/12/13 12:16PM

© Carnegie Learning E In thislesson,youwill: lEArning gOAlS Solving ExponentialEquations What’s y • • • • • • • • SSEntiA equal: Ifa are equal,thenthebasesmustalsobe If the logarithms are equal and the arguments be equal:Iflo g are equal,thenthearguments mustalso If thebasesare equalandthelogarithms lo g The ChangeofBaseFormulastates: is involved. to evaluatelogarithmswhentechnology Change ofBaseFormulamaybenecessary logarithms andnaturallogarithms,sothe Most calculatorscanonlyevaluatecommon rewriting itintermsofadifferent base. calculate anexactvalueforalogarithmby The ChangeofBaseFormulaallowsyouto exponential equations. Analyze different solutionstrategiestosolve log ofbothsides. Solve exponentialequationsbytakingthe Change ofBaseFormula. Solve exponentialequationsusingthe and a,bﬁ1. b (c)5

l id ______lo g 5 lo g

c b a , thenlo g (b) (c) b EAS (a)5lo g

, where a,bc.0 b our Strategy? (a b (c),thena5. ) 5 lo g b (c ). St CO KEy tErm exponential modelsandsolveproblems Construct andcompare linear, quadratic,and Models F-LE Linear, Quadratic,andExponential Build newfunctionsfrom existingfunctions F-BF BuildingFunctions • 4. 5. A Change ofBaseFormula mmO logarithm thesolutiontoa b For exponentialmodels,express asa involving logarithmsandexponents. use thisrelationship tosolveproblems between exponentsandlogarithms (+) Understand theinverserelationship using technology. is 2,10,ore;evaluatethelogarithm c, anddare numbersandthebaseb ndA n COrE rdS fO r mAthEmA St AtE 13.3 ct 5dwhere a, tiCS 941A 17/12/13 12:16PM 13 451451_IM3_TIG_Ch13_917-988.indd 2 941B

Exponential and Logarithmic Equations Chapter 13 misconceptions throughout. solutionmethodsandcommon log of both sidesoftheequation.Studentsanalyzealternative the Change ofBaseFormula.Thenstudentsexplore solvinganexponentialequationbytakingthe Students usethecontextofsubscriberstoanonlinegamesolveexponentialequationsbyusing Overview 17/12/13 12:16PM

© Carnegie Learning Warm Up Match eachlogarithmicequationwithitscorresponding exponentialexpression. lo 1. g logx510 7. logx5e 8. log105x 6. log10052 4. lnx510 5. lo 2. g lo 3. g 2 2 2 (2)5x (4)52 (1024)510 Logarithmic Equation

(a) (f) (b) (g) (c) (h) (e) (d) Exponential Expression 1 g. 0 h. 1 f. 0 d. e. c. 1 b. 0 1 a. 0 2 e 2 2 10 2 x 10

10 2 x

Solving Exponential Equations 13.3

941C 17/12/13 12:16PM 13 13 451451_IM3_TIG_Ch13_917-988.indd 4 941d

Exponential and Logarithmic Equations Chapter 13 17/12/13 12:16PM

© Carnegie Learning World UsageandPopulationStatistics,June30,2012 Internet I In thislesson,youwill: LEArnIng gOALS Solving ExponentialEquations What’s Your Strategy? How long do you think it will take before everyone on Earth isusingtheinternet? onEarth How longdoyouthinkitwilltakebeforeeveryone the Earth’s 7billionpeopleinJuneof2012: • • • world isonline—yet.Here’s usewasgrowingamong aglimpseofhowfastinternet inthe youknowisonline.Butofcoursenoteveryone t mayseemlikeeveryone cai/utai 35,903,569 Oceania/Australia exponential equations. Analyze different solutionstrategiestosolve log ofbothsides. Solve exponentialequationsbytakingthe Change ofBaseFormula. Solve exponentialequationsusingthe North America Latin America Middle East Region Europe Africa Asia ouain(02 nentUes(00 Internet Users (2012) Internet Users (2000) Population (2012) 3,922,066,987 1,073,380,925 593,668,638 348,280,154 223,608,203 820,918,446 Solving Exponential Equations 13.3 108,096,800 105,096,093 114,304,000 18,068,919 KEY TErM 7,620,480 3,284,800 4,514,400 • Change ofBaseFormula 1,076,681,059 254,915,745 273,785,413 518,512,109 167,335,676 24,287,919 90,000,455 13.3 30/11/13 10:15AM 941

17/12/13 12:16PM 941 13 13 451451_IM3_TIG_Ch13_917-988.indd 942 942

• • • • • Questions 1through4 for SharePhase, guiding Questions as aclass. students share theirresponses 4 withapartner. Thenhave parts ofQuestions1through Have studentscompleteall grouping expressions with large numbers. Formula toevaluatelogarithmic along withtheChangeofBase graphing calculatorisused was previously estimated. A is compared toananswer that the ChangeofBaseFormula of theunknownresulting from logarithmic situation.Thevalue used torevisit andevaluate the Base Formulaisintroduced and for unknowns.TheChangeof and usetheequationtosolve after a specified number of days subscribers toanonlinegame calculates thetotalnumberof Students writeanequationthat Problem 1

power of3? Can 400berewritten asa subscribers? it willtaketohave20,000 determine howmanydays What equationisusedto power of3? Can 81berewritten asa 4050 subscribers? it willtaketohave determine howmanydays What equationisusedto the situationP(t)5501 3 Is theequationthatmodels is itP(t)550? 3 Exponential and Logarithmic Equations Chapter 13 t ? t or 13 451450_IM3_Ch13_917-988.indd 942 942 • •

Which answerinthisproblem isexactandwhichansweranestimate? Which twointegerpowersof3doesthevalue400fall between? Problem

Exponential and Logarithmic Equations Chapter 13 1. subscribes willthensend3more peopletosubscribe. On thefirstdayofits release, 50peoplesubscribe.Thecreators estimatethateveryonewho The newestonlinegameisBubblezBurst,ahighlyaddictivethatrunsonsocialmedia. 2. 4. 3. 1 P(t) 550? 3 who willbeplayingBubblezBurstaftertnumberofdays. Write afunctionusingthecreator’s estimatetomodelthetotalnumberofsubscribersP It willtake3daysforBubblezBursttohave4050subscribers. How manydayswillittakeforBubblezBursttohave4050subscribers? I knowthat 3 of 3.InQuestion3,Ihadtouseestimationbecause300isnotapower In Question2,Iwasabletousecommonbasessolvefortbecause81isapower How didyourmethodsinQuestion2and3differ? It willtakeapproximately 5.4daysforBubblezBursttoreach 20,000subscribers. How manydayswillittakeforBubblezBursttoreach 20,000subscribers?

20,000 550? 3 4050 550? 3 P(t) 550? 3 Don’t BurstMyBubble 81 5 3 3 400 5 3 P(t) 550? 3 4 5t 4 5 3 t t t t 5 5243and 3 t t t t 6 5729,so,t6.Iwouldestimatetobeabout5.4. 05/12/13 4:40PM 17/12/13 12:16PM © Carnegie Learning

© Carnegie Learning • • • • • • • Questions 5through7 for SharePhase, guiding Questions • • grouping Is 3 estimation? value resulting from the of BaseFormula,orthe resulting from theChange accurate, thevalue Which valueismore logarithmic expression? used toevaluatethe How isacalculator of BaseFormula? expression usingtheChange evaluate thelogarithmic Is acalculatorneededto equation 4005 3 argument intheexponential Which numberisthe equation 4005 3 logarithm intheexponential Which numberisthe 400 5 3 the exponentialequation Which numberisthebasein responses asaclass. have studentsshare their 7 witha partner. Then Questions 5through Have studentscomplete of BaseFormula. information andtheChange Ask astudenttoread the 5.4 or 3 t ? 5.454 closerto400? t t ? ? 451450_IM3_Ch13_917-988.indd 943

© Carnegie Learning lo g The Change of Base Formulastates: derive it. you willusetheChangeofBaseFormulaandthen logarithm byrewriting itintermsofadifferent base.First, allowsyoutocalculateanexactvaluefora BaseFormula logarithms whosevalueswere notintegers.TheChangeof So far, youhaveusedestimationtodeterminethevalueof 7. 6. 5. b (c)5 400.15¯400 Thecalculatedvalueoflog the originalequation.Whatdoyounotice? value inQuestion6bysubstitutingeachbackinto Compare yourestimateinQuestion3withthecalculated whencheckedintheequivalentexponentialequation. 5.454,whichyields400.15,orapproximately400, exponentialequation. yields377.10whencheckedintheequivalent Theestimatedvalueoflog moreaccurate. log using commonlogs.Roundtothenearest thousandth. Use theChangeofBaseFormulatoevaluatelogarithmicexpression log Question 3asalogarithmicequation. Rewrite theexponentialequationyouwrote in Thecalculatedvalueforlog 377.10¯400 ______ log400 3 3 log3 3 3 5.454 3 3 (400)5x (400)5t 5.454¯x 5.4 ______lo g lo g 3 40053 0400 0400 5x a a (b) (c)

, where a,bc.0andﬁ1 t 3 3 (400)was5.4,which (400)¯5.454is 3 (400)wasapproximately Solving Exponential Equations 13.3 SolvingExponentialEquations 13.3 So, the Change of Base Formula common logs and natural logs. can be helpful to evaluate make that much of a decimal places really logs of other bases. can only evaluate Most calculators difference? Do a few

30/11/13 10:15AM 943

17/12/13 12:17PM 943 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 944 944

• • • • • • • • • • • Questions 8through10 for SharePhase, g share theirresponses asaclass. a partner. Thenhavestudents Questions 8through 10with Have studentscomplete g uiding Questions uiding Questions rouping

314,000,000 subscribers? it willtaketohave determine howmanydays What equationisusedto you know? the values5and6?Howdo Is theargument 600between between thevalues5and6? Why shouldthelogarithmbe two values? fall betweenthesame Does theargument of600 two values? an argument betweenwhich The exponent2.3produces the arguments? argument firstbefore dividing the commonlogofeach Should Tammy havetaken common log? arguments before takingthe Did Tammy dividethe of operationscorrectly? Did Tammy performtheorder 1,000,000 subscribers? will taketohave the numberofdaysit Formula helptodetermine How cantheChangeofBase using apowerof3? Can 20,000berewritten 1,000,000 subscribers? it willtaketohave determine howmanydays What equationisusedto Exponential and Logarithmic Equations Chapter 13

13 451450_IM3_Ch13_917-988.indd 944 944 • •

will taketohave314,000,000subscribers? How cantheChangeofBaseFormulahelptodetermine thenumberofdaysit Can 6,280,000berewritten usingapowerof3?

ExponentialandLogarithmicEquations Chapter 13 9. 8. knowledge ofestimationtoexplainwhyxcouldnotequal2.3. 30,000 subscribers.Describethecalculationerror Tammy made.Then,useyour Tammy wasaskedtoapproximate howmanydaysitwouldtakeBubblezBursttoreach 3 argumentbetween9and27because3 Tammy’ssolutiondoesnotmakesense.Theexponentx¯2.3wouldproducean takingthecommonlogofeachargumentfirst. incorrectly.Shedividedtheargumentsbeforetakingthecommonlog,insteadof Tammy’sreasoningwasincorrectbecausesheperformedtheorderofoperations Itwilltakeapproximately9daystoreachonemillionsubscribers. one millionsubscribers. Use acalculatortodeterminehowmanydaysitwilltakeforBubblez Bursttoreach and theargument600isbetween243and729. 3 Iknowthelogarithmshouldbebetween5and6because3 600 doesnotfallbetween9and27. log ______ log20,000 9.0145428¯x 3 1,000,000550?3 3 (20,000)5x log3 3 20,00053 5x 3 P550?3 x x x Tammy

log ______log 600 log 2005 30,000 550? 3 3 (600)5 log 3 600 5 3 2.3 <

2 59and3

5 x x x x x

3 527,buttheargumentof 5 5243and3 6 5729, 30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning • • • Questions 1through4 for SharePhase, g share theirresponses asaclass. partner. Thenhavestudents Questions 1through 4witha Have studentscomplete grouping the strategytheyusedineach. describe thereasoning behind exponential equationsand last activity, studentssolve exponential equations.Inthe related tosolvingvarious work thatshowseachstep review anddiscussstudent result inequalbases.Students logarithms andequalarguments equal arguments, andequal equal logarithmsresult in equations: equalbasesand strategies tosolvelogarithmic Students usetwodifferent Problem 2 uiding Questions uiding Questions are thebasesequal? If theexponentsare equal, power of2? How can64berewritten asa exponents equal? If the bases are equal, are the 451450_IM3_Ch13_917-988.indd 945

© Carnegie Learning • • • Problem If thelogarithmsare equalandthearguments are equal,are thebasesequal? If thebasesare equalandthelogarithmsare equal,are thearguments equal? How can125berewritten asapowerof5? 4. 1. equations withcommonexponents. Previously yousolvedexponentialequationswithcommonbasesaswell 10. 3. 2. 2 Forlog Explain thestrategyyouusedtosolveeachlogarithmicequationin Question 3. 2 a. Solve eachexponentialequation. BubblezBurst. Itwilltakeapproximately14.2daysforeveryoneintheUnitedStatestosubscribeto how longwouldittakeforeveryoneinthecountrytosubscribeBubblezBurst? In 2012,there were approximately 314millionpeopleintheUnited States.Inthatyear, lo g a. Solve eachlogarithmicequation. equal,Iknowthatthebasesmustalsobeequal. Forlog Iknowthattheargumentsmustalsobeequal. For2 Explain thestrategyyouusedtosolveeachexponentialequationin Question 1. Iknowthatthebasesmustalsobeequal. For I knowthattheexponentsmustalsobeequal.

log w520 2 2 Log ofBothSides y ______ x56 log6,280,000 x x 14.24786588¯x 3 564 2 52 x 3 3 314,000,000550?3 5125,Iconverted125toabaseofpower3.Becausetheexponentsare 564,Iconverted64toapowerofbase2.Becausethebasesare (6,280,000)5x 3 (w)5lo g m 3 (w)5log 3 6,280,00053 (9)5log log3 6 5x 3 P550?3 3 (20) 4 3 (9), (20),becausethebasesare x because x x the logarithms Solving Exponential Equations 13.3 lo g b. b. y m54 y are equalandthelogarithmsare y55 3 3 SolvingExponentialEquations 13.3 5125 55 m (9)5lo g equal 3 and 4 (9) the arguments equal, are equal, equal,

30/11/13 10:15AM 945

17/12/13 12:17PM 945 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 946 946

• • • • • Questions 5through7 for SharePhase, guiding Questions • • grouping

Does lo g Does x5 Does x5lo g common logofthebase? the argument dividedbythe taking thecommonlogof Did Daniellesolveforxby common logofthebase? the argument dividedbythe taking thecommonlogof Did Todd solveforxby responses asaclass. students share their with a partner. Thenhave Questions 5through 7 Have studentscomplete Discuss asaclass. information andformulas. Ask astudenttoread the Exponential and Logarithmic Equations Chapter 13 a (c)5 _____ log c log a a (c)?

? _____ log c log a

? 13 451450_IM3_Ch13_917-988.indd 946 946

ExponentialandLogarithmicEquations Chapter 13 if lo g You justderivedtherelationship that 6.

5. Change ofBaseFormula. You canusethisknowledgetonowderivethe is alsotrue.Ifa5c,thenlo g Consider theexponentialequation propertiesoflogarithmstosolve. Daniellesolvedforxbyfirsttakingthelogofbothsides.Thensheusedthe usedtheChangeofBaseFormula. Toddsolvedforxbyfirstrewritingtheequationasalogarithmicequation.Thenhe Describe howTodd’s andDanielle’s methodsare different . Solvetheexponentialequationforxbytakinglog b. Solvetheexponentialequationfor xbyrewriting itinlogarithmicform. a. a andcare constants. equation 4 ToddTodd andDanielleeachsolvedtheexponential andDanielleeachsolvedtheexponential b (a)5lo g of bothsides. x5log xloga5c log( x x x

2 2 2 4 a x5 x 2 1 1 1 b x Todd x x21 a )5logc (c),thena5.Theconverse 1

b (a)5lo g b (c). a x 5c,where xistheunknowninexponentand byrewriting itinlogarithmicform the equation balanced. of an equation keeps log of both sides Taking the ( x 21)log4550 log ( 4 Danielle the balanced. x sides methods to remove the 215 keeps 4 used two different x21 unknown from the x21 x x ) 5log50 550 <3.822 5 exponent. exponent. You ______log 50 log 50 log 4 log 4 just

30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning 7. log(8 c. Solve theexponentialequation 8 methods .Roundtothenearest thousandthandcheckyourwork. xlog85log38.96 Iknowthatx5log Bothmethodsyieldthesameresult. How do the results from these two methods 8 x x5 x¯1.761 x )5log38.96 538.96 ______log38.96 log8 a (c)andx5 38.94¯38.96 Check:8 8 x¯1.761 x5 x5log x 538.96 x 538.96usingboth Todd’s andDanielle’s ______log38.96 _____ loga logc Solving Exponential Equations 13.3 log8 8 (38.96) 1.761 ,thereforelog 038.96

demonstrate the Change of Base Formula? SolvingExponentialEquations 13.3 a it’s important to isolate the term with the variable first (c)5 when solving equations, Remember that before solving. _____ logc loga .

30/11/13 10:15AM 947

17/12/13 12:17PM 947 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 948 948

• • • • Questions 8 for Share Phase, g share theirresponses asaclass. partner. Thenhavestudents Questions 8and9witha Have studentscomplete g uiding Questions uiding Questions rouping

power ofthebase? if theequationisequaltoa Whose methodwillonlywork each other? set theexponentsequalto of equivalentbasesandthen equation byusingtheidea Which studentsolvedthe of bothsides? equation bytakingthelog Which studentsolvedthe of BaseFormula? and thenusingtheChange a logarithmicequationfirst equation byconvertingitinto Which studentsolvedthe Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 948 948

ExponentialandLogarithmicEquations Chapter 13 8 . John, Bobbi,andRandyeachsolvedtheequation 9 Willeachmethodworkforeverylogarithmicequation?Describeanylimitationsof b. Describeeachmethodused. a. Randy’smethodwillworkonlyifthebasesare John’smethodandBobbi’smethodwilleachworkforeveryexponentialequation. each method. exponentsequaltoeachother. Randysolvedtheequationbyusingtheideaofequivalentbasesandsettingthe Bobbisolvedtheequationbytakingthelogofbothsides. usedtheChangeofBaseFormula. Johnconvertedtheexponentialequationintoalogarithmicequationandthen x x x 1 1 1 9 x 1 4 4 4 John x 4

5 5 5 5 5

27 1.5 lo g 2 ____ log 27 log 9 2.5 9

(27)

2(x 2x ( 3 randy 3 3 1 2 2 1 9 (x ) 4) 2x x x 1 8 1 1 4) x 4 4

5 5 5 5 5 5 5

3 3 3

3 2 27 2 2.5 3 3 5

( x

equal. log ( 9 1 x14 Bobbi 4)log9 527. x x

x 1 1 9

1 x 4 4 4 1 x ) 4

5 5 5 5 5 5 log27 27 log27 1.5

_____ log 27 2 log 9 2.5

30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning • • • • Question 9 for Share Phase, guiding Questions value oftheunknown? determine theapproximate Change ofBaseFormulato Did bothstudentsusethe base e? logarithmic equationwith Which studentwrote a base 10? logarithmic equationwith Which studentwrote a using acalculator? base whichcanbeevaluated logarithmic expression with a the exponentialequationtoa use amethodthatchanged Did bothAmeetandNeha 451450_IM3_Ch13_917-988.indd 949

© Carnegie Learning 9 . method iscorrect . Describe thesimilaritiesanddifferences intheirmethods.Explainwhyeachstudent’s Ameet andNehaeachtookthelogarithmofbothsidestosolve 24 AmeetandNeha’smethodsare BaseFormulaanddeterminedanapproximatevalueof0.506. Eachstudent’smethodiscorrectbecausetheyevaluatedlog base Theirmethodsare equationwithabasewhichcanbeevaluatedusingacalculator. e ,whereasNehawroteanequivalentlogarithmicexpressionwithbase10. Check: 2 4 ln (2 4 4.99 x 0.506 ln 24 2 4 Ameet 2 4 x

< 0 0 x ) x x x

5 5 5 5 5 5 < 5 ln 5 5 In 5 differentinthatAmeetwroteanequivalentlogarithmicexpressionwith ____ ln 24 0.506 ln 5

similarinthattheybothrewrote24 Solving Exponential Equations 13.3

SolvingExponentialEquations 13.3 2 4 Check: log (2 4 x 4.99 <5 0.506 log245 2 4 neha x 05 05 2 4 x x x ) 5log x 5 5 <0.506 24 _____ log 24 5usingtheChangeof log 5 x x

5. 5asalogarithmic

30/11/13 10:15AM 949

17/12/13 12:17PM 949 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 950 950

• • • • • • • Questions 10and11 for Share Phase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents Questions 10and11witha Have studentscomplete grouping

exponent associatedwith before sheworkedwiththe power of3? Can 27berewritten asa in thissituation? both sidesoftheequation Can youtakethelogof power of Can 36berewritten asa this situation? of BaseFormulain Can youusetheChange power of4? Can 21berewritten asa Did Ashleymultiply9by of operationscorrectly? Did Ashleyperformtheorder Exponential and Logarithmic Equations Chapter 13 __ 2 3

__ 3 1

? 13 451450_IM3_Ch13_917-988.indd 950 950

ExponentialandLogarithmicEquations Chapter 13 11. 10 1 . 4 a. Solve eachexponentialequation.Explainwhyyouchosethemethodthatused x¯20.2325 togetherbeforeworkingwiththeexponenton 3 3 Ashleydidnotperformtheorderofoperationscorrectlyandmultiplied9and Describe theerror inAshley’s reasoning .Thensolvetheequationcorrectly . x5 2 222x5log 3 2 2 (3) ( x5 I x235 x235log __ 1 3 2x 2x 2x5log used x23 ) 4 4 2x x23 515 515 515 2516 x¯5.20 521 ______ log15 the log3 ______log21 log21 log4 log4 3 3 2 (15) (15)2 Change 4 22 (21) 13 log ( 3 2x log35 9 of

Ashley (

__ Base 3 1

2x 5 2x

2x 52.464973521. ) 2x 2x )5 x < Formula

5 5 log15

log15

1.232 15 15 _____ log 15 log 3 because

__ 1 3 . 21 cannot be written asa power __ 3 1 of 4. 30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning Be prepared toshare yoursolutionsandmethods. 2? 3 c. 10? b. 3 2?3 Itookthelogofbothsidesbecause36cannotbewrittenasapowerof 3 Iusedcommonbasesbecause27canberewrittenas3 ( __ 2 3 2x5log x5 2x5 ) 2x x¯4.42 536 3 2?3 5x 5x (

__ 2 3 1555 11555

) 2x ( ______log log36 3 3 3 3 5x3 __ 2 1 5360 5x 5x 5x x5 ) 554 3 53 527 __ 3 2 ______log log36 ( (36) __ 2 3 __ 3 5 ) ( 3 __ 2 3 ) Solving Exponential Equations 13.3 SolvingExponentialEquations 13.3 3 . __ 2 3 .

30/11/13 10:15AM 951

17/12/13 12:17PM 951 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 952 952

Check forStudents’Understanding

Exponential and Logarithmic Equations Chapter 13 2. Solve eachexponentialequation.Explainwhyyouchosethemethodthatused. 1. 7?(2) 120couldnotberewrittenasapowerof2,soIusedtheChangeofBaseFormula. 7 ?(2 ) 24couldnotberewrittenasapowerof2,soIusedtheChangeofBaseFormula. 2 2 2 x x 25 25 x2 5¯4.58 x2 55 x2 55log (2) 16530 16530 2 2 3x5log 3x56.90689... 3x5 x 3x 3x 3x x¯2.30 25 5840 x¯9.58 5840 5120 524 ______log120 ______log24 log2 log2 2 (120) 2 (24) 17/12/13 12:17PM

© Carnegie Learning E In thislesson,youwill: lEArning gOAlS Solving l l • • • • • • • • SSEntiA ogging On sides, andevaluatingthepower. exponents, takingthelogarithmofboth using commonbasesor include: takingthe Strategies forsolvingexponentialequations converting toexponentialform. setting basesorarguments equal,and include: theuseofproperties oflogarithms, Strategies forsolvinglogarithmicequations the logarithmisunknown. Change ofBaseFormulawhenthevalue logarithmic equationbyapplyingthe It ismore advantageoustosolveasimple unknown orthebaseisunknown. exponential formwhentheargument is logarithmic equationbyrewriting itin It ismore advantageoustosolveasimple logarithmic equations. Complete adecisiontree todetermineefficient methodsforsolvingexponentialand Solve logarithmicequationsarisingfrom real-world situations. Solve logarithmicequationsusingproperties. Solve forthebase,argument, andexponentoflogarithmicequations. l id EAS ogarithmic Equations n th root ofbothsides, St CO exponential modelsandsolveproblems Construct andcompare linear, quadratic,and Models F-LE Linear, Quadratic,andExponential Build newfunctionsfrom existingfunctions F-BF BuildingFunctions 4. 5. A mmO logarithm thesolutiontoa b For exponentialmodels,express asa involving logarithmsandexponents. use thisrelationship tosolveproblems between exponentsandlogarithms (+) Understand theinverserelationship using technology. is 2,10,ore;evaluatethelogarithm c, anddare numbersandthebaseb ndA n COrE rdS fO r mAthEmA St AtE 13.4 ct 5dwhere a, tiCS 953A 17/12/13 12:17PM 13 451451_IM3_TIG_Ch13_917-988.indd 2 953B

Exponential and Logarithmic Equations Chapter 13 and logarithmic equation. contexts. A‘decisiontree’ iscreated describingthefirststepusedtosolveeachtypeofexponential equations containingmultiplelogarithms,andstudentssolvelogarithmicinreal-world exponential equationsorusingtheChangeofBaseFormula.Properties oflogarithmsare usedtosolve Students solvelogarithmicequationsforthebase,argument, orexponentbyrewriting themas Overview 17/12/13 12:17PM

© Carnegie Learning Warm Up 4. Use theproperties oflogarithms tosolveeachofthefollowing. 2. 3. 1. 2log1122log55log Rewrite 2log115usingasinglelogarithmandevaluate. log Rewrite lo g log Rewrite lo g log Rewrite lo g 5 2 3 (8 (7)5 ( ___ 5 x x y 4 4 6 ) ) 5log 5log _____ log5 log7 2 3 5 (7)usingcommonlogarithmsandevaluate. (8 (

___ 5 x x y 4 3 ¯1.209 2 4 6

(8)1log (5)16log ) usingmultiplelogarithms.

) usingmultiplelogarithms. ¯0.6848 5log4.84 ( 3 ___ 11 (x) 5 2 2 (y) 24log 2 ) 2 (x) Solving Logarithmic Equations 13.4

953C 17/12/13 12:17PM 13 13 451451_IM3_TIG_Ch13_917-988.indd 4 953d

Exponential and Logarithmic Equations Chapter 13 17/12/13 12:17PM

© Carnegie Learning H In thislesson,youwill: LEArnIng gOALS Solving LogarithmicEquations Logging On Can yourecognizethedifferencebetweenastrongandweakpassword? Avoid usingthesamepasswordformultiplesites. • Include numbers,symbols,andcapitallowercaselettersifallowedby • • Aminimumpasswordlengthof12to14characters,ifpermitted. • follows somecommonguidelines: somakesureyourpassword Passwords aretheretoprotectyourprivateinformation, complexity, andunpredictability. average, toguessitcorrectly. Thestrengthofapasswordisfunctionitslength, trials anattackerwhodoesnothavedirectaccesstothepasswordwouldneed,on itisanestimateofhowmany attacks.Initsusualform, guessing andbrute-force • • • • the system. relativeorpetnames,otherbiographicalinformation. usernames, Avoid, words,letterornumberpatterns passwordsthatuserepetition,dictionary logarithmic equations. Complete adecisiontree todetermineefficient methodsforsolvingexponential and Solve logarithmicequationsarisingfrom real-world situations. Solve logarithmicequationsusingproperties . Solve forthebase,argument, andexponentoflogarithmicequations. Password strengthisameasureoftheeffectivenesspasswordinresisting ave youeverbeenaskedtochangeyourpassword? Solving Logarithmic Equations 13.4 13.4 30/11/13 10:15AM 953

17/12/13 12:17PM 953 13 13 451451_IM3_TIG_Ch13_917-988.indd 954 954

• Questions 1and2 for SharePhase, g • • grouping simple logarithmicequations. the bestmethodtosolveseveral the equation.Studentschoose Change ofBaseFormulatosolve more advantageoustoapplythe of thelogarithmisunknown,it is unknown.Andwhenthevalue argument isunknownorthebase it inexponentialformwhenthe logarithmic equationbyrewriting advantageous tosolveasimple conclude thatitismore the baseisunknown.Students the exponentisunknown,and when theargument isunknown, equations twodifferent ways a chartbysolvinglogarithmic a baby crying. Students complete and theintensityofsound the decibellevelforcitytraffic, the decibel levelofaquietlibrary, solve forunknowns,suchas Students usetheequationto by alogarithmicequation. or loudnessofsound,ismodeled The formulaforthedecibellevel, Problem uiding Questions uiding Questions

a quietlibrary? determine thedecibellevelof What equationisusedto a class. share theirresponses as partner. Thenhavestudents Questions 1through 3witha Have studentscomplete Discuss asaclass. information andformula. Ask astudenttoread the Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 954 954 • • • • • • •

What happenstothequantities How isthenumber500,000,000usedinequation? Is thebase,argument, orexponentunknowninthissituation? What equationisusedtodeterminethedecibellevel ofcity traffic? What happenstothequantities How isthenumber1000usedinequation? Is thebase,argument, orexponentunknowninthissituation? Problem

ExponentialandLogarithmicEquations Chapter 13 of asoundisgivenby A decibelisaunitusedtomeasure theloudnessofsound.Theformulafor 1. than thethreshold sound. that canbarely beperceived .ThequantityIisthenumberoftimesmore intenseasoundis 2. where dB is the decibel level . Thequantity 1 dB510log Aquietlibraryhasadecibellevelof30. threshold sound:I51000 The soundinaquietlibraryisabout1000timesasintensethe dB510log Citytraffichasadecibellevelofapproximately87. the threshold sound.Calculatethedecibellevelforcitytraffic . The soundoftraffic onacitystreet iscalculatedtobe500milliontimesasintense

530 510log1000 510log ¯86.9 510log500,000,000 510log Keep ItDown! ( ( ( ( __ __ ______ ______I I 1000 500,000,000 I I 0 0

) ) I 0 I 0 I ) 0 I 0 I inthenumeratoranddenominatoroflog? 0 I I inthenumeratoranddenominator ofthelog? 0 0 .Calculatethedecibellevelofaquietlibrary . ) dB 510log I 0 is the intensity of the threshold sound—a sound (

__ I I 0

)

30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning • • • Question 3 for Share Phase, guiding Questions the equation? 115 decibels usedin How isthevalueof this situation? exponent unknownin Is thebase,argument, or sound ofacryingbaby? determine theintensityof What equationisusedto 451450_IM3_Ch13_917-988.indd 955

© Carnegie Learning 3. in yourcalculations.Methodswithfewerstepscanbemore accurate andefficient . steps involved.Themore stepsthere are, themore chancesyouhavetointroduce anerror When solvingequationswithacertainmethod,itisimportanttoconsiderthenumberof • • logarithmic equation: Previously, youdevelopedtwostrategiestosolveforanyunknowninasimple argument, and exponent. unknown inasimplelogarithmicequation,youconsidertherelationship betweenthebase, Each equationyouwrote inQuestions1through 3isalogarithmicequation. To solveforan Apply theChangeofBaseFormula. Rewrite thelogarithmicequationasanexponential. thresholdsound. Thesoundofacryingbabyismorethan300,000,000,000timesasintenseasthe this soundthanthethreshold sound? The soundofacryingbabyregisters at115decibels.Howmanytimesmore intenseis 10 11.55log 115510log dB510log 11.5 I¯3.16310 5 __ I I 0

( __ I I 0

) ( ( __ __ I I I I 0 0

) ) 11 I 0 Solving Logarithmic Equations 13.4 SolvingLogarithmicEquations 13.4

30/11/13 10:15AM 955

17/12/13 12:17PM 955 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 956 956

• • • • • • • • • • • Question 4 for Share Phase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents Questions 4through 7with a Have studentscomplete grouping

unknown inthissituation? Is thebase,log,orargument lo g the argument intheexample Which numberrepresents lo g the loginexample Which numberrepresents lo g the base intheexample Which numberrepresents this situation? Change ofBaseFormulain exponential equationorthe Is iteasiertousethe exponential equation? the valueresulting from the Change ofBaseFormula,or the valueresulting from the Which valueismore accurate, Change ofBaseFormula? expression afterusingthe evaluate thelogarithmic Is acalculatorneededto unknown inthissituation? Is thebase,log,orargument l og the argument intheexample Which numberrepresents l og the loginexample Which numberrepresents l og the base intheexample Which numberrepresents Exponential and Logarithmic Equations Chapter 13 8 8 8 5 5 5 (145)5x? (145)5x? (145)5x? (x) 5 3.1? (x)53.1? (x)53.1? 13 451450_IM3_Ch13_917-988.indd 956 956 • • • • • •

this situation? Is iteasiertousetheexponential equationortheChange ofBaseFormulain Is thebase,log,orargument unknowninthissituation? Which numberrepresents theargument intheexamplelo g Which numberrepresents theloginexamplelo g Which numberrepresents thebaseinexamplelo g this situation? Is iteasiertousetheexponentialequationorChange ofBaseFormulain

ExponentialandLogarithmicEquations Chapter 13 Argument Is Exponent Is Unknown Unknown Unknown Base Is 4. Solve eachlogarithmicequationusingtwodifferent methods. lo g lo g lo g Example x 8 (24)56.7 5 (145)5x (x)53.1 equation. Then solve for x. First rewrite as an exponential log(8 xlog85log145 ( x 146.827¯x 6.7 8 8 x x x¯2.3933 x5 ) x )5log145 ___ 6.7 6.7 5145 1 x¯1.6069 5 5 5(24) 524 3.1 ______log145 log8 5x ___ 6.7 1 x (24)56.7? x Formula. Then solve for x. First apply the Change of Base (24)56.7? 1.6069¯x 0.2060¯logx 10 ______ log24 log2456.7logx x logx 146.827¯x (24)56.7? ______ 0.2060 log145 10 2.3933¯ x log8 _____ log5 logx logx¯2.1668 2.1668 56.7 ¯x 5 x ¯x 53.1 30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning • • • Questions 5through7 for SharePhase, guiding Questions preferred method? as anexponentialequationa why isrewriting theequation When thebaseisunknown, preferred method? Change ofBaseFormulaa unknown, why is applying the When theexponentis a preferred method? exponential equation the equationasan unknown, whyisrewriting When theargument is 451450_IM3_Ch13_917-988.indd 957

7. . . exponentialformwhentheargumentisunknownorthebaseisunknown. Itismoreadvantageoustosolveasimplelogarithmicequationbyrewritingitin exponential form? When mightitbemore efficient tosolveasimplelogarithmicequationby rewriting itin usingacalculator. method.You Whentheexponentisunknown,rewritingasanexponentialequationismypreferred preferredmethodbecausetheexpressioncanbeevaluatedusingacalculator. Whentheexponentisunknown,applyingtheChangeofBaseFormulaismy methodbecausetheexpressioncanbeevaluatedusingacalculator. Whentheargumentisunknown,rewritingasanexponentialequationismypreferred Circle yourpreferred methodforsolving.Explainyourchoice . Consider thepositionofunknownforeachlogarithmicequationinQuestion4. ChangeofBaseFormulawhenthevalueofthelogarithmisunknown. Itismoreadvantageoustosolveasimplelogarithmicequationbyapplyingthe Change ofBaseFormula? When mightitbemore efficient tosolveasimplelogarithmicequationbyapplyingthe cantakethe n th rootofeachsidetosolvefortheunknownandevaluate Solving Logarithmic Equations 13.4 SolvingLogarithmicEquations 13.4

30/11/13 10:15AM 957

17/12/13 12:17PM 957 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 958 958

• • • Question 8 for Share Phase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents parts ofQuestion8witha Have studentscompleteall grouping

which methodispreferred? If theexponentisunknown, is preferred? unknown, whichmethod If thebaseorargument is unknown inthissituation? Is thebase,log,orargument Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 958 958

ExponentialandLogarithmicEquations Chapter 13 8 . lo g a. applying theChangeofBaseFormula.Explainyourchoice exponential equations.Boxthethatcanbesolvedmore efficiently by Circle thelogarithmicequationsthatcanbesolvedmore efficiently when rewritten as In(x14)53.8 e. lo g g. lo g c. isthepreferredmethod. rewritingasanexponentialequation Becausetheargumentisunknown, isthepreferredmethod. rewritingasanexponentialequation Becausetheargumentisunknown, isthepreferredmethod. rewritingasanexponentialequation Becausethebaseisunknown, isthepreferredmethod. rewritingasanexponentialequation Becausethebaseisunknown, x12 4 12x (x13)5 (7.1)53 (8)514.7 __ 2 1

lo g b. lo g f. log(42x)51.3 h. lo g d. isthepreferredmethod. applyingtheChangeofBaseFormula Becausetheexponentisunknown, isthepreferredmethod. rewritingasanexponentialequation Becausetheargumentisunknown, isthepreferredmethod. applyingtheChangeofBaseFormula Becausetheexponentisunknown, isthepreferredmethod. applyingtheChangeofBaseFormula Becausetheexponentisunknown, 3 4 .5 11 (4.6)52x (12)5x27 (9)5x21 30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning • • • Question 9 for Share Phase, guiding Questions responses asaclass. Then havestudentsshare their Question 9withapartner. Have studentscomplete grouping with thisquadraticequation? Which factorsare associated equation factorable? Is thequadratic to onesideoftheequation? Do youmovealloftheterms 451450_IM3_Ch13_917-988.indd 959

© Carnegie Learning 9 . lo g a. Solve eachlogarithmicequation.Checkyourwork lo g b. 6 6 165 2 2 05 05(x22)(13) 65 x52,23 05(x28)(12) 05 1 x58,2 4 5 5 2 6 ( ( x x x x x x x x 2 2 2 2 2 2 2 2 26x)54 1x)5 1x 1x26 1x 26x 26x16 26x Solving Logarithmic Equations 13.4 Check:log Check:log SolvingLogarithmicEquations 13.4 log log 6 ( 2 (23) ( (22) 2 6 2 (2 (8 1 (23) log log 2 26(2) 2 2 log log 26?8)04 12)01 6 6 (6)51 (6)51 2 2 (16)54 (16)54 ) 01 ) 04

30/11/13 10:15AM 959

17/12/13 12:17PM 959 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 960 960

• • • Questions 1and2 for SharePhase, g • • grouping multiple logarithms. paths andsolveequationswith to identifyerrors insolution Students review studentwork the equationtoafamiliarform. multiple logarithmstorewrite logarithmic equationscontaining of logarithmsare usedin the startofsurgery. Properties needs tobeadministered at the amountofmedicinethat medication leavesthebody, and body, therateatwhich of medicinetoremain inthe it takesforacertainamount unknowns suchasthetime use theequationtosolvefor logarithmic equation.Students body ismodeledbya medication leftinapatient’s The formulafortheamountof Problem 2 uiding Questions uiding Questions

the equation? How is20%usedin in theequation? How isthevalue10mgused patient’s body? the medicinetoremain inthe time itwilltakefor2mgof determine theamountof What equationisusedto a class. share theirresponses as partner. Thenhavestudents Questions 1through 3witha Have studentscomplete Discuss asaclass. information andformula. Ask astudenttoread the Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 960 960 • • • • • • •

What strategywasusedto solvethisproblem? The decimal0.206isequivalent towhatpercent? How is5mgusedintheequation? How is20mgusedintheequation? How isthevalue6hoursusedinequation? patient’s body? What equationisusedtodeterminerateatwhichthe medicineisleavingthe What strategywasusedtosolvethisproblem? Problem

ExponentialandLogarithmicEquations Chapter 13 2. patient’s bodycanbepredicted bytheformula long amedicationwilllastwhentheywriteprescriptions . While mostmedicationshaveguidelinesfordosageamounts,doctorsmustalsodeterminehow 1. milligrams, andristherateatwhichmedicineleavesbody . medicine leftinthepatient’s bodyinmilligrams,Aistheoriginaldoseofmedicine where tisthetimeinhourssincemedicinewasadministered, Cisthecurrent amountof 2 Themedicineisleavingthebodyatarateofabout20.6%perhour. patient’s body .Atwhatrateisthemedicineleavingbody? Six hoursafteradministeringa20-milligramdoseofmedicine,5milligramsremain ina t¯7.2 5 log(12r)¯0.100 t5 patient’sbody. Itwilltakeabout7.2hoursfor2milligramsofthemedicinetoremaininthe patient’s body? 20% perhour .Howlongwillittakefor2milligramsofthemedicineto remain inthe A patientisgiven10milligramsofmedicinewhichleavesthebodyatrate

10 Doctor, Doctor! ______log(12r) log(120.20) log 20.100 log 65 r¯0.206 r¯ 12 ( t5 ¯12r __ C A ( ___ 10 2 ) ______log(12r) log(12r) ) log log 10 ( ( ___ 20 __ C 5 A 20.100 ) ) t

______

log (12r log (

__ C A

) )

The amount of medicine left in a Theamountofmedicineleftina 30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning • • • • • Question 3 for Share Phase, guiding Questions solve thisproblem? What strategywasusedto the equation? How is15%usedin the equation? How is4mgusedin used intheequation? How isthevalue8hours of surgery? administered atthestart of medicinethatmustbe to determinetheamount What equationisused 451450_IM3_Ch13_917-988.indd 961

© Carnegie Learning 3 . 10 8-hoursurgery. Thepatientshouldreceiveabout14.7milligramsofmedicinebeforethe 15% perhour, howmuchmedicinemustbeadministered atthestartofsurgery? patient’s bodyattheendofsurgery, andthemedicineleavesbodyatrateof A patientisundergoing an8-hoursurgery .If4milligramsof medicine mustbeleftinthe log 20.5646 ( __ A 4 A¯14.68 A¯ 85 t5 ) ¯20.5646 ¯ ______10 ______A ______log(120.15) log(12r) 4 log 20.5646 4 log ( __ C A ( __ A 4 ) ) Solving Logarithmic Equations 13.4 SolvingLogarithmicEquations 13.4

30/11/13 10:15AM 961

17/12/13 12:17PM 961 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 962 962

• • • • • • • Questions 4and5 for SharePhase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents Questions 4and5witha Have studentscomplete grouping

solve thisequation? What strategycanyouuseto logarithms inthissituation? you usetocombinethe What ruleorproperty can multiple logs? to solvetheequationwith Which methodwasused form tosolvetheequation? equation intoexponential followed byconvertingthe Power RuleofLogarithms Which studentusedthe the leftsideofequation? combine thelogarithmson Did Lorenzo correctly the leftsideofequation? combine thelogarithmson Did Santiagocorrectly the leftsideofequation? combine thelogarithmson Did Georgia correctly Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 962 962

ExponentialandLogarithmicEquations Chapter 13 4. logarithms torewrite theequationinaformyoualready knowhowtosolve. If alogarithmicequationinvolvesmultiplelogarithms,youcanusetheproperties of Solvelog51x2.Checkyourwork b. Analyze Georgia’s Explaintheerror ineachstudent’s reasoning . a. to log51logx52 side Georgia, write log 5 log (5 of log(5x)52 georgia the 1 log5 Santiago, 10055x 1 log 10 205x equation. 1 0

95 2 x 55x ) x 2 1

, Santiago’s 5 5 5 5 log

2 2 5 x 1 and x

Each as x log 5 Lorenzo log(5x , andLorenzo’s work. Lorenzo student 1 5 ) logx and incorrectly 1 should x then x 5 5 5 Check:log51log2002 converted 2

2 log 51 2 have combined 3 Santiago used to 5 x x x the 52 52 5 the log10052 exponential Power __ 5 2 logarithms

Rule form on of Logarithms to the solve. left 30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning 5 . lo g a. Solve eachlogarithmicequation.Checkyourwork lo g b. log log 5 2 2 5 (45x)2lo g (8)13lo g (45x)2log (8)1log log log 2 (8 5 2 ( ( 2 ____ 45x 6458 x x 5 (x)56 5 2 2 (3)51 3 3 3 (3)5 1 85 x5 2 6 ) ) 5 5 58 56 56 __ 3 1 5515x 1 ) 5x 515x 51 x 3 x x 3 3 Solving Logarithmic Equations 13.4 Check:log Check:log SolvingLogarithmicEquations 13.4 number, the base y variables for the logarithmic equation, and not equal to 1, and the argument = l og 2 5 (8)1log ( b 45?

must be greater than zero. x the restrictions on the log . The variable log __ 3 1 2 (8 ?2 Don’t forget ) 2log 2 2 (64)06 b (2 must be greater than 0 log 64564 2 3 3 6 ) ) 06 06 064 5 5 (5)51 (3)01 y can be any real

30/11/13 10:15AM x 963

13 17/12/13 12:17PM 963 13 13 451451_IM3_TIG_Ch13_917-988.indd 964 964

• • • • • • • Questions 6and7 for SharePhase, guiding Questions share theirresponses asaclass. partner. Thenhavestudents Questions 6through 9with a Have studentscomplete grouping

you know? this situation?Howdo Is there anextraneouslogin equation factorable? Is thequadratic solve thisequation? What strategycanyouuseto in thissituation? you usetocombinethelogs What ruleorproperty can multiple logarithms? to solvetheequationwith Which methodcanyouuse values ofx? does thisaffect possible involve xor original logarithmicequation Does theargument ofthe Kate’s solution? between Pippa’s solutionand What isthedifference Exponential and Logarithmic Equations Chapter 13 x 2 ? How 13 451450_IM3_Ch13_917-988.indd 964 964

ExponentialandLogarithmicEquations Chapter 13 ? 6. 7. becausetheargumentis Kateiscorrect.Both210and10are Who iscorrect? Explainyourreasoning . lo g Pippa andKatedisagree aboutthesolutiontologarithmicequation Check:log log 275 05(x27)(15) 05 Solve lo g log x57,25 3 3 lo g Kate 3 5 5 5 3 3

( (x24)1log ( 7, ( x x x x x x 2 2 x 2 log 22x35 2 2lo g 2 ) log 2 2 2 22x8 22x8)53 22x8 2 5 5 7) 3 5 (100)2log lo g (x24)1lo g lo g (10 5 ( 5 5

5 x 452. (

100 2 2 (4) __ x 4 25 ) ) 2 5 2log 2log

log x ) 2

3 5 5 5 5 5 5 (x12)53 2 2 10,

x 5 __ __ 25525 x

4 4 5 (25)02 2 2 2 5 5 5

(4)02 (4)02 (4)02 3 2 (x12)53.Checkyourwork 2 025 10 x 2 ,whichisalwayspositive. Reject must argument ofalogarithm lo g Pippa solutionsoftheoriginallogarithmicequation 5 ( x be greaterthanzero. log log 527 27 3 extraneouslogundefinedfor 25 log Check:log lo log log 2 2 ) 5 10 becausethe ( 2 ) (210) 5 (100)2log lo g lo g 2 3 5 3 (724)1log ) 2log (2513)log 5

( (4) 100

x 25 4 5 2 5 5

(25)02

x 2 g 5 ) 5 25525 (4)02 (4)02

3 (3)1log 5 5 5 5 5 5 2 025

2 2 x

10, __ __

x x 4 4 3 2 2 2 (712)03

© Carnegie Learning • • • • • Questions 8and9 for SharePhase, guiding Questions solve thisequation? What strategycanyouuseto in thissituation? you usetocombinethelogs What ruleorproperty can multiple logarithms? to solvetheequationwith Which methodcanyouuse exponential form? then convertedto of theequation,and logarithms ontheleftside applied theproperties of of theequationequalto0, Which studentsetoneside equal toeachother? each equivalentlogarithm then setthearguments of the rightsideofequation, properties oflogarithmson Which studentappliedthe 451450_IM3_Ch13_917-988.indd 965

© Carnegie Learning Check: 2 log 6 Elijah Explain whyeachstudent’s methodiscorrect . 8. 9. logarithmsontheleftsideoftheequation,andthenconvertedtoexponentialformtosolve. Zanderiscorrectbecausehesetonesideoftheequationequalto0,appliedthepropertiesof equation log ( 6 2 log 6 36 Elijah 72 Elijah andZandereachsolvedthelogarithmicequation2log65x. 2log(x11)53) a. Solve eachlogarithmicequation.Checkyourwork 2 log ( 6 ) is log 36 5 5 5 5 log((x11) correct 2log(x11)5loglog(3) x and

x log x log 2 __ 2 x 2 12x15

)

0 5 0 ( (x11) then

__ 2 x 2 log 72 log 36 log

because )

log 2 set (

2 __ 15x 72 2 ) 2 5log(x(13)) 5x(13) 2

the )

log 2 x he 2 13x arguments applied the of

properties each Solving Logarithmic Equations 13.4 equivalent Check: 2log6072 log ( 6 of Check:2 logarithms Zander SolvingLogarithmicEquations 13.4 2 ) 2log log ( 6 log 365 log logarithm (

2 ___ 36 ) 0log x log

) log 1log250 1log250 on log((2) 2log65 (1 equal the log45log4 ( 1 0 1

( __ 72

x __ 72 x 2 15

1) right

0 )

50 5 572 ) ) to

0log(1(4)) 0 log each __ __ 72 72 log1 x side x

2 other of 1 log2 the log to (1 solve. 1 3)

30/11/13 10:15AM 965

17/12/13 12:17PM 965 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 966 966

Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 966 966

ExponentialandLogarithmicEquations Chapter 13 In(x23)12)5(224) c. 2log(x3)541 b. 2lo g d. log x In((x23)(2))5In(2124) In(x23)1In(2)5In(224) x59,2 (x29)(12)50 x x 2log (x26)(4)50 2 2 2 log((x23) 2log(x23)5log41log 27x1850 25x165224 210x12450 x 3 2 (x 9, 26x1954x15 9)( 3 3 x5lo g x x5log 2 x58 (x23) 2 2 ) x 564 5log 2 1 2 x54,6 2 ) 3 3 3 5log(4x215) 54x215 41lo g 41log (4(16)) 3 3 16 16 (

x 2 ( x2 ___ 15 4

___ 15

) 4

) Check:2 In extraneousInundefinedfor2 Check: Check:2 (9 2 3) 2 In In log 2 log log log 1 In6 (2(2 log log ((2 log(1 log(3 3 3 In ( (6 (4 2) 3 3 8 64 8 8 2) 2 log95log9 log10log1 (9 ) 2 2 1 In7 In 2 3) 0 5 0 0 1 24) 2 2) 3) 3) 2 2 log log log log )0log )0log 42 0 0 1 3 3 3 3 0 0 5 In log4 log4 4 In 64 64 64 In In 1 ((2 ( ( (2(9) 42 42 4 4 log 1 1 ( 2) ( __ 9 4 __ 1 4 ) 3 log log 2 2) ) ) 1 ) 16 24) ( ( 62 42 0 ______15 15 4 4 ) ) 30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning • • • • talk the talk for SharePhase, guiding Questions responses asaclass. Then havestudentsshare their decision tree withapartner. Have studentscompletethe grouping each equation. students wouldusetosolve and describesthefirststep solving eachtypeofequation advantageous strategyfor which demonstratesthemost to completea‘decisiontree’ provided. Studentsusethe list students havesolvedis and logarithmicequation of eachtypeexponential A listcontaininganexample talk thetalk nth root ofbothsides? associated withtakingthe first stepinthesolution Which examplehasa exponential form? associated withconvertingto first stepinthesolution Which examplehasa arguments equal? bases equalorsettingthe associated withsettingthe first stepinthesolution Which examplehasa the properties oflogarithms? associated withtheuseof first stepinthesolution Which examplehasa 451450_IM3_Ch13_917-988.indd 967

© Carnegie Learning • • Talk theTalk the power? Which examplehasafirststepinthesolutionassociated withevaluating bases orcommonexponents? Which examplehasafirststepinthesolutionassociated withusingcommon • • • • • • • • you havesolved. The listshowncontainsanexampleofeachtypeexponentialandlogarithmicequation have hadtoconsiderthestructure andcharacteristicsofeachequation. You havesolvedavarietyofexponentialandlogarithmicequations.Whendoingso,you 1. lo g x log x125243 5 lo g 4 lo g 2 5 .5

x 5 .1 x12 517 Describethefirststepyouwouldusetosolveeachequation. b. Foreachbranchofthedecisiontree, writetheappropriate exampleequationfrom a. advantageous strategytosolveeachtypeofequation. Complete thedecisiontree onthefollowingpagestodemonstratemost 522 5x 8 3 6 532 (x)50.3 (x)2lo g (x17)5lo g Answerswillvary. Seedecisiontree. the list. 3 (9)5 6 31 Solving Logarithmic Equations 13.4 SolvingLogarithmicEquations 13.4

30/11/13 10:15AM 967

17/12/13 12:17PM 967 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 968 968

Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 968 968

1 log log Example: logarithms. Usethepropertiesof st

step: multiple log5 x ExponentialandLogarithmicEquations Chapter 13 24 1 2 log2 log3 5 log Example: setargumentsequal. Setbasesequalor 1 st step: 6 ( x 1 7) Solving LogarithmicEquations 5log location ofthelogarithms. log 5 Identify thenumberand 6 (31) log Example: 1 exponentialform. Convertto st step: 8 log 5constant (x)50.3 . multiple log5constant 1 log Example: logarithms. Usethepropertiesof st step: 3 (x ) 2 log 3 (9) 5 5 30/11/13 10:15AM 17/12/13 12:17PM © Carnegie Learning

© Carnegie Learning bothsides. x Example: Takethe 1 5.5 st step: 522 base n th rootof Be prepared toshare yoursolutionsandmethods. 2 2 Example: exponents. basesorcommon Usecommon 1 st x12 step: 532 Solving ExponentialEquations Identify thelocation of theunknown of theunknown. exponent Solving Logarithmic Equations 13.4 1 5 5 Example: bothsides. Takethelogarithmof st x 517 step: . SolvingLogarithmicEquations 13.4 4 4 Example: Evaluatethepower. 1 st 5.1 step: 5x value

30/11/13 10:15AM 969

17/12/13 12:17PM 969 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 970 970

Check forStudents’Understanding

Exponential and Logarithmic Equations Chapter 13 The firststepinthesolutionpathisto: Create anexampleproblem tofiteachfirststep. 3. 2. 4. 1. 2 2 Answerswillvary. Use commonbasesorexponents. 2 2 Answerswillvary. Evaluate thepower. log Answerswillvary. Convert toexponentialform. 8 8 Answerswillvary. Take thelogarithmofbothsides. x 6.4 x 5510 23 5x 5 5128 (x)568 17/12/13 12:17PM

© Carnegie Learning E In thislesson,youwill: lEArning gOAlS and l Applications ofExponential So WhenWill i • • • SSEntiA used tomodelsituationsinthereal world. Exponential andlogarithmicequationsare Use logarithmicmodelstoanalyzeproblem situations. Use exponentialmodelstoanalyzeproblem situations. ogarithmic Equations l id EAS Uset exponential modelsandsolveproblems Construct andcompare linear, quadratic,and Models F-LE Linear, Quadratic,andExponential Build newfunctionsfrom existingfunctions F-BF BuildingFunctions St CO 4. 5. A mmO logarithm thesolutiontoa b For exponential models,express asa involving logarithmsandexponents. use thisrelationship tosolveproblems between exponentsandlogarithms (+) Understand theinverserelationship using technology. is 2,10,ore;evaluatethelogarithm c, anddare numbersandthebaseb ndA his? n COrE rdS fO r mAthEmA St AtE 13.5 ct 5dwhere a, tiCS 971A 17/12/13 12:18PM 13 451451_IM3_TIG_Ch13_917-988.indd 2 971B

Exponential and Logarithmic Equations Chapter 13 elapsed time sinceapersondiedinhours. the growth ofonlinefanbases, andusingaformuladerivedfrom Newton’s Lawtodeterminethe would grow bestinacertain typeofsoil,usingthecontinuouspopulationgrowth formulatoproject equations andlogarithmicinclude:usingthepHformulatoanalyzetypeofcrops that Exponential andlogarithmicequationsare usedtomodelsituations.Applicationsofexponential Overview 17/12/13 12:18PM

© Carnegie Learning Warm Up more intensethanthethreshold sound? It takesasoundofapproximately 45decibelstowakeasleepingperson.Thissoundishowmanytimes perceived, andIisthenumberoftimesmore intensethesoundisthanthreshold sound. The formulafortheloudnessofasoundisgivenby 10 thresholdsound. Thesoundittakestowakeasleepingpersonisabout31,623timesmoreintensethanthe where dBisthedecibellevel, 4.55log dB510log 45510log 4.5 I531,622.7766 5 ( __ I I 0 ) ( __ I I 0 ) ( ( __ __ I I I I 0 0 ) ) I 0 I 0 istheintensityofthreshold sound—asoundthatcanbarely be dB 510log Applications of Exponential and Logarithmic Equations 13.5 (

__ I I 0

)

971C 17/12/13 12:18PM 13 13 451451_IM3_TIG_Ch13_917-988.indd 4 971d

Exponential and Logarithmic Equations Chapter 13 17/12/13 12:18PM

© Carnegie Learning temperature forthefirsthourafterdeathandthen1degreeeachthat. body ofthumbtoestimatethetimedeath:subtract2degreesfromnormal rule officialswhoareresponsibleforverifyingdeaths—oftenusea Coroners—government the environment. tothetemperaturedifferencebetweenobjectand at aratethatisproportional what’s knownasNewton’s LawofCooling,whichstatesthatanobjectcoolsdown In thislesson,youwill: LEArnIng gOALS and LogarithmicEquations Applications ofExponential So WhenWill IUseThis? C • • Use logarithmicmodelstoanalyzeproblem situations. Use exponentialmodelstoanalyzeproblem situations. based ontwotemperaturereadingsofthebody. Specifically, investigatorsuse rime investigatorsuselogarithmicequationstoestimateabody’s timeofdeath Applications of Exponential and Logarithmic Equations 13.5 13.5 30/11/13 10:15AM 971

17/12/13 12:18PM 971 13 13 451451_IM3_TIG_Ch13_917-988.indd 972 972

• • Questions 1through4 for SharePhase, g • • grouping is feasible. a farmer’s choiceofcrops growing onions,anddecide if optimal hydrogen ionrange for grow, determinethesoil’s list thecrops afarmershould students usethisinformationto and vegetablesare given and soil levelsforavarietyoffruits and limewater. OptimalpH of hydrogen ionsinvinegar soda, andtheconcentration level oforangejuice,baking unknowns suchasthepH use theequationtosolvefor logarithmic equation.Students a pHlevelismodeledby The formulaforcalculating Problem 1 uiding Questions uiding Questions

this situation? exponent unknownin Is thebase,argument, or determine thesolution? What equationisusedto a class. share theirresponses as partner. Thenhavestudents Questions 1through 4witha Have studentscomplete Discuss asaclass. information andformula. Ask astudenttoread the Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 972 972

Problem

ExponentialandLogarithmicEquations Chapter 13 2. 3. determined bytheconcentrationofhydrogen ions.TheformulaforpHis The pHscaleisaformeasuringtheacidityoralkalinityofsubstance,which where H 1. with apHof7are neutral.Forexample,plainwaterhasapHof7 value lessthan7are acidic.SolutionswithapHvaluegreater than7are alkaline.Solutions 1 pH52log(5.012310 The concentrationofhydrogen ionsinbakingsodais5.01231 0 ThepHlevelislessthan7,sotheorangejuiceisacidic. ThepHleveloforangejuiceisabout3.7. pH<3.701 pH52log0.000199 10 Theconcentrationofhydrogenionsinvinegarisabout0.0063molepercubicliter. hydrogen ionsinvinegar . Vinegar hasapHof2.2.Determinetheconcentration ThepHlevelisgreaterthan7,sobakingsodaisalkaline. ThepHlevelofbakingsodaisabout8.3. pH<8.3 H pH52logH Determine thepHlevelofbakingsoda,andthenstatewhetheritisacidicoralkaline. The H H pH52logH level oforangejuice,andthenstatewhetheritisacidicoralkaline.

H 2.252logH H pH52logH 22.2 H H 1 Efficient pHarming 1 istheconcentrationofhydrogen ionsinmolespercubicliter .SolutionswithapH 5H <0.0063 1 concentrationinorangejuiceis0.000199molepercubicliter .DeterminethepH 1

1 1

29 ) pH 52log H 1

© Carnegie Learning • • • • • • Questions 5through7 for SharePhase, guiding Questions • • grouping were feasible? determine whichcrops What equationisusedto this situation? exponent unknownin Is thebase,argument, or growing onions? hydrogen ionrangefor determine thesoil’s optimal What equationsare usedto of crops? soil usedtodeterminethelist How wasthepHlevelof this situation? exponent unknownin Is thebase,argument, or the soil? determine thepHlevelof What equationisusedto a class. share theirresponses as partner. Thenhavestudents Questions 5through 7with a Have studentscomplete chart asaclass. level ofsoil.Discussthe information aboutthepH Ask astudenttoread the 451450_IM3_Ch13_917-988.indd 973

© Carnegie Learning • • • • Can thefarmerplantpeanuts? Can thefarmerplantgarlic? Why won’t beetsgrow inthefarmer’s garden? Is thebase,argument, orexponentunknowninthissituation? The chartshowstheoptimalpHsoillevelsforavarietyoffruitsandvegetables. Generally, thesoiltendstobeacidicinmoistclimatesandalkalinedry. Different typesofplantsandvegetablesrequire varyingdegrees ofsoilacidity . The pHlevelofsoilisusedtodeterminewhichplantswillgrow bestinanarea . 5. 4. 10 cubicliter. Theconcentrationofhydrogenionsinlimewaterisabout0.000000000001moleper spinach,sweetcorn,andyamsinthissoil. Thefarmercangrowavocados,beets,onions,peppers,potatoes,raspberries, pH¯6.2 A farmermeasures thehydrogen ionconcentrationofthesoiltobe631 0 Lime waterhasapHof12.Determinetheconcentrationhydrogen ionsinlimewater . pH52log(6310 cubic liter .Listthecrops thatthefarmercangrow inthis typeofsoil. H pH52logH H 212 H 1252logH 1 Applications of Exponential and Logarithmic Equations 13.5 5H 50.000000000001 Mushrooms Asparagus 5 .027 6 .028 7 .028 5 .026 6 .527 Peppers Spinach Carrots 1

1 1

27 ) ApplicationsofExponentialandLogarithmicEquations 13.5 Sweet Corn Avocados Potatoes 6 .027 5 .826.5 6 .22.8 5 .026 6 .027 Onions Garlic Raspberries 6 .028 6 .02.5 5 .026 6 .527.0 5 .626 Peanuts Lettuce Beets Yams 27 moleper

30/11/13 10:15AM 973

17/12/13 12:18PM 973 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 974 974

Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 974 974

ExponentialandLogarithmicEquations Chapter 13 7. 6. H 0.000000158¯H 0.000000631¯H concentration ofthesoiltobe3.16 10 The farmerwantstoplantspinach,carrots, andbeets.Shemeasures thehydrogen ion OnionsgrowbestinasoilwithpHlevelsbetween6.2and6.8. Theoptimalrangeofhydrogenionconcentrationinthesoilis1.58310 growing onions. Determine therangeofsoil’s optimalhydrogen ionconcentrationfor 6.31310 pH¯5.5 pH52log(0.00000316) peanutsinstead. Spinachandcarrotswillgrowinthissoil,butbeetswillnot.Shecanplantgarlicor Explain yourreasoning . 10 10 H 26.85 logH 26.25 logH 27 H 6.852logH H 6.252logH 26.2 26.8 molepercubicliter. 5 logH 5 logH 1 1

1 1 1 1

1 1

© Carnegie Learning • • • Questions 1and2 for SharePhase, guiding Questions a class. share theirresponses as partner. Thenhavestudents Questions 1through 6witha Have studentscomplete grouping talk showhost. pro-football runningback, anda a pro-football quarterback, a Fan basesincludeaboyband, the numberoffollowersgrows. of followers,ortherateatwhich takes togetaspecifiednumber amount oftime,howlongit followers afteraspecified calculate thenumberofonline exponential equationsthat model tocreate additional equation. Studentsusethe is modeledbyanexponential continuous populationgrowth The formulaforcalculating Problem 2 in days? growth represents thetime for continuouspopulation Which variableinthemodel of growth? growth represents the rate for continuouspopulation Which variableinthemodel number offollowers? growth represents theinitial for continuouspopulation Which variableinthemodel 451450_IM3_Ch13_917-988.indd 975

© Carnegie Learning • • • • • • Problem How is100,000usedinGina’s formula? Is thebase,argument, orexponentunknowninthissituation? How manymonthsare from MaytoSeptember? How is26%usedinGina’s equation? How is18,450usedinGina’s equation? number offollowersaftertmonths? Which variableinthemodelforcontinuouspopulation growth represents the 1. the monthlyincrease inthenumberoffollowersonherclients’socialmediasites. Gina, asocialmediamanager, usesthemodelforcontinuouspopulationgrowth toproject 2. 2 IfGinastartedworkingwiththebandonMay1st,howmanyonlinefollowersdid b. Howlongdidittakefortheband tosurpass c. Recall thattheformulaforcontinuouspopulationgrowth isN(t)5 UseGina’s claimtowriteafunctionrepresent thenumberofonlinefollowersthat a. 18,450 onlinefollowersandshewasabletoincrease theirfollowersby26%permonth. Gina claimsthatwhenshestartedworkingwithanup-and-comingboyband, theyhad • • • • quantity represents intermsofthisproblem situation.

N(t)518,450 Thereare Thebandwouldhavehad52,199onlinefollowersbySeptember1st. they havebySeptember1st? N(4)518,450 In(5.420054201)<0.26t November15th. afterapproximately6.5months,orapproximately Thebandsurpassed100,000onlinefollowers 100,000 onlinefollowers? ¯52199.054 N(t)518,450 the boybandhadaftertmonths. N(t)isthenumberoffollowersaftermonths. tisthetimeinmonths ristherateofgrowth N Follow Me! Applications of Exponential and Logarithmic Equations 13.5 0 istheinitialnumberoffollowers 5.420054201< 1.690105816<0.26t 4monthsfromMaytoSeptember,sot54. 100,000518,450 e e e 6.500¯t 0.26t 0.26t (0.26 ? 4) ApplicationsofExponentialandLogarithmicEquations 13.5 e 0.26t e 0.26t approximately very end! Instead, use the “ANS” feature on your calculator to recall the previous step. your decimals until the N Don’t round 0 e rt .Statewhateach

30/11/13 10:15AM 975

17/12/13 12:18PM 975 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 976 976

• • • • • • • Question 3 for Share Phase, guiding Questions

Gina’s equation? How isitusedin What is105,326doubled? band’s equation? similar totheboy quarterback’s equation How isthepro-football band’s equation? different from theboy quarterback’s equation How isthepro-football this situation? or exponentunknownin Is thebase,argument, Gina’s equation? How is4125usedin Gina’s equation? How is105,326usedin this situation? exponent unknownin Is thebase,argument, or Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 976 976

ExponentialandLogarithmicEquations Chapter 13 3. Write anexponentialfunctiontorepresent thenumberofonlinefollowers b. Ifhehad4125onlinefollowerswhenhired Gina,atwhatmonthlyratedidhis a. to 105,326. she beganmanaginghisaccountthree yearsago,hisonlinefollowershavegrown Gina’s topclientisapro footballquarterback.Onherwebsite,Ginaclaimsthatsince Howlongwouldittakethequarterback todoublehiscurrent onlinefollowing c. 0.6931471806¯0.09t In(25.53357576)¯36r N(t)5105,326 continues togrow atthesamerate. quarterback willhaveinanygivenmonth,assumingthenumberoffollowers Hisnumberofonlinefollowersgrewatapproximately9%permonth. number ofonlinefollowersgrow? currentonlinefollowing. Itwouldtakeapproximately7.7months,or7monthsand21days,todoublehis of 105,326? 0.0899998412¯r 25.53357576¯ 3.239994283¯36r 210,6525105326 105,32654125 7.702¯t In250.09t N(t)510,5326 N(t)5 25 e 0.09 e 0.09t t N e 36r 0 e rt e e 36r e 0.09t 0.09t 30/11/13 10:15AM 17/12/13 12:18PM © Carnegie Learning

© Carnegie Learning • • • • • • • • • • • • • Questions 4through6 for SharePhase, guiding Questions this situation? exponent unknownin Is thebase,argument, or would haveinoneyear? followers thetalkshowhost to predict thenumberof What equationwasused it changetheanswer? the nearest hundredth, would If the rate was not rounded to rounded off? Was therateJoshused checking Gina’s work? What ratedidJoshusewhen for thetalkshowhost? of increase onlinefollowing determine theratepermonth What formulaisusedto the quarterback? online followersthan running backwillhavemore determine whetherthe What equationisusedto this situation? exponent unknownin Is thebase,argument, or had whenhehired Gina? followers therunningback determine howmanyonline What equationisusedto in theequation? How is15monthsused the equation? How is14%usedin the equation? How is62,100usedin quarterback’s equation? different from thepro-football running back’s equation How isthepro-football 451450_IM3_Ch13_917-988.indd 977

© Carnegie Learning 6 4 5 . . . Ginaclaimsthatshecantriplethe host’s numberofonlinefollowersin6months. a. 5200 onlinefollowers. Gina acquires atalkshowhostasnewclient.Thecurrently has Therunningbackhad7605followerswhenhehiredGina15monthsago. him whenhehired Gina15monthsago? followers .Ifhehasseena14%increase inhisfollowers,howmanypeoplefollowed Gina alsomanagesarunningbackonthesameteamwhocurrently has62,100online willhavethesamenumberoffollowersasthequarterback. Yes.Afterapproximately10.6months,or10monthsand18days,therunningback quarterback? Ifso,when?not,explainyourreasoning . Will therunningbackeverhavesamenumberofonlinefollowersas 7604.544¯ (1.696070853) In(1.696070853)¯0.05t ______ 62,100 62,1005 e Ginaisclaimingapproximately18%permonthgrowthinonlinefollowers. Determine theratepermonthofincreased onlinefollowing. 0.1831020481¯r Applications of Exponential and Logarithmic Equations 13.5 0.14?15 1.098612289¯6r 1.696070853¯ 105,326 5 15,60055200 N N N In356r 0 0 0 e e e 0.09t 0.09t 0.14?15 35 t¯10.566 562,100 ¯ e e e 6r 0.05t 0.14t ApplicationsofExponentialandLogarithmicEquations 13.5 e 6r e 0.14t

30/11/13 10:15AM 977

17/12/13 12:18PM 977 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 978 978

Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 978 978

ExponentialandLogarithmicEquations Chapter 13 AssumingthatGina’s promised rateofgrowth istrue,project howmanyfollowers c. Whatistheerror inJosh’s method?CheckGina’s workcorrectly . JoshdecidestocheckGina’s work. b. N(6)55200 N(12)55200 Thetalkshowhostwillhaveapproximately46,800followersinayear. the talkshowhostwillhaveinayear . ¯15,600 Joshusedtheroundedrateof0.18ratherthanamoreprecisedecimalvalue. ¯46,800 N(t)55200 N(t)55200 N N Gina didnottriplethehost’sfollowers. 15312 ﬁ3(5200) N N N N (6) ¯15312.33367 (6) 55200 ( t t ) 55200 Josh e e 0.1831020481t e e 0.1831020481(6) 0.1831020481t 0.1831020481(12) e e 0.18 0.18(6) t

© Carnegie Learning • • • • • • Question 1 for Share Phase, guiding Questions • • grouping and theroom temperature. temperature ofthebody the timeofdeathgiven use themodeltocalculate logarithmic equation.Students has diedismodeledbya elapsed timesinceaperson The formulaforcalculatingthe Problem 3 this situation? exponent unknownin Is thebase,argument, or denominator intheformula? to determinethevalueof What informationisneeded formula determined? denominator inthe How isthevalueof numerator intheformula? to determinethevalueof What informationisneeded formula determined? numerator inthe How isthevalueof this situation? estimate thetimeofdeathin What equationisusedto a class. share theirresponses as partner. Thenhavestudents Questions 1and2witha Have studentscomplete Discuss asaclass. information andformula. Ask astudenttoread the 451450_IM3_Ch13_917-988.indd 979

to calculatetheelapsedtimesinceapersonhasdied.Theformulais A coroner usesaformuladerivedfrom Newton’s LawofCooling,ageneralcoolingprinciple, where Tisthebody’s measured temperature in8F, Ristheconstantroom temperature in8F, 1. and tistheelapsedtimesincedeathinhours. 3 b. and averagingthecalculatedtimesofdeath. A more accurateestimateofthetimedeathisfoundbytakingtwo ormore readings a. temperature oftheroom where thebodywasfoundis708F . At 8:30

t5210In At 9:30 t¯6hoursand8minutes t¯6.13 t¯210In0.542 t¯7hoursand58minutes t¯7.96 t¯210In0.451 t5210In t5210In 6hoursand8minutesearlier,or2:52 estimatedtimeofdeathisapproximately 7 At 9:00 Using to estimatethetimeofdeath. Usingthe9:00 to estimatethetimeofdeath. t5210In Cracking theCase Applications of Exponential and Logarithmic Equations 13.5 hours am the acoroner wascalledtothehomeofapersonwhohaddied.Theconstant am am and thebody’s measured temperature was82 .9 thebody’s measured temperature was85 .5 9:30 ( ( ( ( ______ ______ ______ ______98.6270 85.5270 98.62R 82.9270 98.62R 98.6270 58 T2R T2R am am minutes body oytmeaue the temperature, body t 5210In ApplicationsofExponentialandLogarithmicEquations 13.5 ) ) ) ) temperature, earlier, or (

______98 .62R T 2R 1:32 the am am . estimated

. )

time the surrounding temperature is 8 8 constant room temperature. If F . F . not constant, then there is a Usethisbodytemperature Usethisbodytemperature the formula assumes a more more involved formula. of death involved Notice that is approximately formula.

30/11/13 10:15AM 979

17/12/13 12:18PM 979 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 980 980

Exponential and Logarithmic Equations Chapter 13 13 451450_IM3_Ch13_917-988.indd 980 980

ExponentialandLogarithmicEquations Chapter 13 Whenwillthetemperature ofthebodydrop to608F?Explainyourreasoning . e. Compare theestimatedtimeofdeathfrom part(a)and(b).Are youranswersfairly c. , Assumingthebodyremains inaroom withaconstanttemperature of708F, d. 5 Theargumentisnegative,sothesolutionisundefined. t5210ln 608F,becausetheconstantroomtemperatureis708F. Assumingthebodyisnotmoved,thetemperatureofthebodywillnotdropto 0.091¯ Amoreaccurateestimatedtimeofdeathis2:12 each Yes. close? Determineamore accurateestimatedtimeofdeath. After24hours,thetemperatureofthebodywillbe72.68F. determine thetemperature ofthebodyafter24hours. 1:32 or40 minutesaftertheearliestestimatedtimeofdeath. Ideterminedthetimehalfwaybetweenthetwocalculatedtimesofdeath, 22.45In 72.6¯T 2.60¯T270 e e 22.4 245210In 210ln The t5210In am other. 5 140minutes52:12 ______T270 ______T270 estimated 28.6 28.6 ( ( ( ______ _____ ______98.6270 28.6 210 T270 60270 28.6 ( ( ) ______ ______98.6270 98.62R T270 T2R ) times ) of ) ) death am are fairly close— am within . 80 minutes of 30/11/13 10:15AM 17/12/13 12:18PM © Carnegie Learning

© Carnegie Learning • • • • • • • • • • • • • Question 2 for Share Phase, guiding Questions of death? the twoestimatedtimes What ishalfwaybetween estimated timesofdeath? How closeare thetwo 7:00 of deathwithrespect tothe determine theestimatedtime What equationisusedto 6:30 of deathwithrespect tothe determine theestimatedtime What equationisusedto than 70°? the bodyeverbelower 70°, canthetemperature of If theroom temperature is exponential equation? equation berewritten asan Can thelogarithmic solve thisproblem? What strategyisusedto problem situation? it represent inthe this situation?Whatdoes What isthevalueofRin the formula? How is24hoursusedin times ofdeath? between thetwoestimated What timeishalfway each other? death withinanhourof Are theestimatedtimesof from theprevious situation? How isthissituationdifferent this situation? estimate thetimeofdeathin What equationisusedto pm pm bodytemperature? bodytemperature? 451450_IM3_Ch13_917-988.indd 981

Usingthe6:30 5:25 calculatedtimesofdeath,or7.5minutesaftertheearliestestimatedtimeofdeath. moreaccuratetimeofdeath,Ideterminedthetimehalfwaybetweenthetwo Theestimatedtimesofdeathare t¯1hourand35minutes t¯1.58 t¯210In0.854 Usingthe7:00 t¯50minutes t¯0.834 t¯210In0.920 At 6:00 t5210In t5210In 1hourand35 minutesearlier,or5:25 50minutesearlier,or5:40 At 6:30 found atthebottomofstairsinahallway, where theconstanttemperature is658F . Yes.Ibelieveherstorybecausetheaverageestimatedtimeofdeathis5:33 Explainyourreasoning. forensic evidence? immediately calledforanambulance.Isherstatementconsistentwiththe A witnessclaimsthatthepersonfelldownstairsaround 5:30pm,andshe measured temperature was93.78F . t5210In t5210In Applications of Exponential and Logarithmic Equations 13.5 pm 17.5minutes¯5:33 pm pm , acoroner wascalledtothehomeofapersonwhohaddied.Thebody thebody’s measured temperature was95.98F, ( ( ( ( ______ ______ ______ 98.6265 93.7265 98.62R 98.6265 95.9265 98.62R T2R T2R pm pm bodytemperature,theestimatedtimeofdeathisapproximately bodytemperature,theestimatedtimeofdeathisapproximately ) ) ) ) ApplicationsofExponentialandLogarithmicEquations 13.5 pm . pm within15minutesofeachother.To pm . andat7:00 pm determinea thebody’s pm .

30/11/13 10:15AM 981

17/12/13 12:18PM 981 13 13 13 451451_IM3_TIG_Ch13_917-988.indd 982 982

Check forStudents’Understanding

Exponential and Logarithmic Equations Chapter 13 t5 How oldisthecartonearest tenthofayear? depreciation forthiscaris 12%peryear. A carwasoriginallypurchased for$32,000andiscurrently valuedat$20,000.Theaveragerateof C istheoriginalvalueofitem,andryearlyratedepreciation expressed asadecimal. Thecarisabout3.7yearsold. t¯3.7 t5 the formulat5 Some itemssuchasautomobilesare worthlessovertime.Theageofanitemcanbepredicted using ______log(120.12) log(12r) log log ( ______32,000 20,000 ( __ C V ) ______log(1 2r) ) log (

__ C V

)

, where tistheageofiteminyear, V isthevalueofitemaftertyears, 17/12/13 12:18PM

© Carnegie Learning Chapter Key TermS 13.1 13.1 • • • and Argument (13.2) Logarithm withSameBase Zero Property ofLogarithms(13.2) logarithmic equation(13.1) Logarithmic equation: log log Solve thelogarithmicequation. Example need torewrite bothsidesoftheexponentialequationsothattheyhavesameexponent. the exponentialequationsothattheyhavesamebase.Ifbaseisunknown,youwill make itsimplertosolve.Iftheexponentisunknown,youwillneedrewrite bothsidesof argument isunknown,writingtheequationinexponentialformwillisolateargument and To solvealogarithmicequation,beginbyconvertingittoanexponentialequation.Ifthe Solving Logarithmiceq Exponential equation: 3 logarithmic equation. Arrange theterms3,4,and81tocreate atrueexponentialequationand Example b exponential equations. the relationship between the base,argument, andexponentoflogarithmic A logarithmicfunctionistheinverseofanexponentialfunction.Thediagrambelowshows Writing ex ase n ( exponent  3

25  13 n n n n n 5

______

3 3 3 3 2 2 2 2 ) 5

5 5 5( 5 5 25 5 5 argument  __ 3 2 3 Summary __ 3 2 ponential andLogarithmiceq

25 2

 __ 3 1 ) __ 1 3

4 5 81 3 (81) 54 ⇔

uations l og base (argument) 5 • • • • Change ofBaseFormula(13.3) Power RuleofLogarithms(13.2) Quotient RuleofLogarithms(13.2) Product RuleofLogarithms (13.2) exponent uations 09/12/1317/12/13 11:4112:18AMPM 983 13 13 451450_IM3_Ch13_917-988.indd451451_IM3_TIG_Ch13_917-988.indd 984 984 984

13.2 13.2 13.1 Exponential and Logarithmic EquationsExponential andLogarithmicEquations Chapter 13 log 4 log Write thelogarithmicexpression 4 log Example To writealogarithmicexpression asasinglelogarithm,usetheproperties oflogarithms. Writing LogarithmicExpressions asaSingleLogarithm Write thelogarithmicexpression log Example To writealogarithmicexpression inexpandedform,usetheproperties oflogarithms. Writing LogarithmicExpressions inExpandedForm Estimate thevalueof log Example the logarithmofupperlimit. closer totheupperlimit,valueofgivenlogarithmwillalsobe also beclosertothevalueoflogarithmlowerlimit.Similarly, iftheargument is argument .Iftheargument isclosertothelowerlimit,valueofgivenlogarithmwill than thegivenargument andtheclosestlogarithmwhoseargument isgreater thanthegiven To estimatethevalueofalogarithm,identifyclosestlogarithmwhoseargument isless Estimating Logarithms 5 log Therefore, log Because 210iscloserto216than36,theapproximation shouldbecloserto3than2. 2 , 3 ,n log 6 3 (36), log

6 (

(x) 1 2 log ___ 5 2 x y 5 2

) 5 log 6 6 (210)<2.9. (210), log 6 3 ( (5) 1 2 log y) 27 log 6 (210)tothenearest tenth. 6 (216) 6 3 (x)5 log (x)2 log 3

(

6 ___

2 5 6 6 (x) 1 2 log 3

y x (2) 2 5 log ( (

5 2 ______

x x y

) 4 3 2 x inexpanded form.

) 7 y

)

6 ( 3 ( y) 27 log y) 6 (x)asasinglelogarithm. 30/11/13 10:15AM

© Carnegie Learning 13.3 13.3 quotient .Finally, solveforthevariable. the argument andthecommonlogofbase.Useyourcalculatortodeterminethat equation usinglogarithms.Next,writethelogarithmasquotientofcommonlog a, bﬁ1. To solveanexponentialequationusingtheChangeofBaseFormula,firstwrite x ¯8.086 x 25¯3.086 x ¯2.223 3x¯6.669 3x 24¯.669 The ChangeofBaseFormulastatesthat log of BaseFormula Solving ExponentialEquationsUsingtheChange 6 Solve theexponentialequationbytakinglogofbothsides. Example the commonlogs.Then,solveforvariable term on one side . Next, take the common log of both sides . Use your calculator to determine To solveanexponentialequationbytakingthelogofbothsides,firstisolatevariable Solving ExponentialEquationsbyTaking theLogofBothSides 8 Solve theexponentialequationbyusingChangeofBaseFormula. Example x 25 (x 25)log65252 log( 6 6 3x 245 3x24 x25 5257 17 5 259 x25 x25 ______log 257 log 8 ) 5log252 5252 ______log 252

log 6

b (c)5 ______log log a a (b) (c)

, where a,bc.0and Summary Summary Chapter 13 Summary Chapter 13

30/11/1317/12/13 10:1512:18AMPM 985 13 13 451450_IM3_Ch13_917-988.indd451451_IM3_TIG_Ch13_917-988.indd 986 986 986

13.4 Exponential and Logarithmic EquationsExponential andLogarithmicEquations Chapter 13 log Solve thelogarithmicequation.Checkyouranswer . Example using theproperties oflogarithms .Then,solve To solvealogarithmicequation,rewrite theequationusingexponentsorrewrite theequation Solving LogarithmicEquations Check: 29isanextraneoussolution. 6 x1 log log (x 19)(24)50 x log x 529,4 log 2 15x2360 6 6 (x15))52 ( 6 41 log 6 x x (x15)52 2 2 15x)2 15x 6 log 6 log 5 (41 5)02 6 (4(9)) 02 2 6 (36) 02 2 9,

36 5 6 2 4 036 30/11/13 10:15AM

© Carnegie Learning 13.5 The vehicleisabout7yearsold. 6.999997787¯t 2 log 0.4474042553¯2.0499t10 the answer . problems, identifywhatthe questionisaskingandthenusethegivenmodeltodetermine Some problem situations canbemodeledwithexponentialequations. To solvethese Using ExponentialModelstoAnalyzeProblemSituations Given V547,000( 10 model toanswerthequestion. The givenexponentialequationmodelstheappraisedvalueofavehicleovertime.Use Example V 547,000( 10 vehicle isifitsappraisedvalue$21,028. 0.4474042553¯ 10 21,028547,000( 10 log 0.4474042553¯ 10 0 .3492998896¯2.0499t 20 .0499t 20 .0499t ), where trepresents thetimeinyears,determinehowold 20 .0499t 20 .0499t 20 .0499t ) ) Summary Summary Chapter 13 Summary Chapter 13

30/11/1317/12/13 10:1512:18AMPM 987 13 13 451450_IM3_Ch13_917-988.indd451451_IM3_TIG_Ch13_917-988.indd 988 988 988

13.5 Exponential and Logarithmic EquationsExponential andLogarithmicEquations Chapter 13 5 A littlemore than205monthlypaymentswouldneedtobemadepayoff theloan. ¯205.3630583 mortgage canbedeterminedbytheformulan5 The numberofmonthlypaymentsapersonmustmaketorepay acarloanorhome Example the answer . problems, identifywhatthe questionisaskingandthenusethegivenmodeltodetermine Some problem situations canbemodeledwithlogarithmicequations. To solvethese Using LogarithmicModelstoAnalyzeProblemSituations ¯ n 5 $79,000 mortgageloanat4.2%interest . Calculate thenumberofpayments$540permonththatwouldneedtobemadeona interest rateperpayment periodindecimalform. original valueoftheloan,prepresents theamountofmonthlypayment,andIis ¯ ¯

______2log ______2log 0 .0015173768 2log (10.512037037) 0 .3116131403 2log 0.487962963 log(1 1I log 1.0035 log (110.0035) ( ( log

1 2 1 2 (

1 ______(

I ?P

______0 .042 ) p 12

______

0 .042

12 ) 540

) (79,000)

)

______2log log (11I (

1 2 _____ I ?P p )

)

, where P 0 represents the 30/11/13 10:15AM