Links in learning logarithms

Rachael Kenney and Signe Kastberg Purdue University, USA

he beautiful history of the development of logarithms (Smith & Confrey, T1994), coupled with the power of the logarithmic function to model various situations and solve practical problems, makes the continued effort to support students’ understanding of logarithms as critical today as it was when slide rules and logarithmic tables were commonly used for computation. They continue to play an important role despite the fact that calculators are now used for many computations involving logarithms: logarithmic scales can increase the range over which numbers can be viewed in a meaningful way. As described in the senior secondary curriculum, logarithmic scales are used regularly in astronomy, chemistry, acoustics, seismology, and engineering and students should be able to “identify contexts suitable for modelling by logarithmic functions” and be able to “use logarithmic functions to solve practical problems (Australian Curriculum and Assessment Reporting Authority [ACARA], 2012) By the time students enter a mathematics course at University, they have likely been exposed to logarithms in Year 10 as an enrichment topic and logarithmic functions in Senior Secondary Mathematics. While the preparation for University study is slightly different in the United States, concerns regarding student struggle with understanding is consistent. We have repeatedly witnessed students’ difficulties with logarithms while teaching university mathematics courses. Our observations mirror those of other teachers. For example, Hurwitz (1999) notes that, “Students often have difficulty thinking of a logarithm as the output of a function because the notation used for logarithms does not look like the familiar f(x) notation” (p. 334). Gramble (2005) relays that teachers often tell students that logarithms are exponents, but, “for some reason students hear the terms exponents and logarithms but often do not understand the relationship between them” (p. 66). Hurwitz’s statement can be applied to other functions such as f(x) = sin(x), and f(x) = a(x) that are included in the Mathematical Methods curriculum and which differ from earlier functions encountered; however, students’ struggles with logarithms and logarithmic functions are the focus here. Exploration Australian Senior Mathematics Journal vol. 27 no. 1

12 Links in learning logarithms Australian Senior Mathematics Journal vol. 27 no. 1 13 ) (x For b ), where ), is defined (x ≠ 1, if and b

(called the (1) = 0 for all b = log = y > 0, b (ACARA, 2011). =1. As a function, log a As =1. 0 b > 0 and b 25 = 2 has identical meaning identical has 2 = 25 5 )) for x (x b = 25 is true, we can also say that the logarithm = 25 is true, we can also say that the 2 ) = log . These ideas can be used to build understanding (x x ) = b is the power to which a given number b is the power to which a given number (x –1 ) (or similarly f (e.g., see Larson, Edwards & Hostetler, 2006). It is clear that a (e.g., see Larson, Edwards & Hostetler, y (x b = b = log y . Recognising the relationship that results from the composition of inverse composition of the from results that relationship the Recognising . > 0, is not just an exponent (or number) but is a function and, as such, has > 0, is not just an exponent (or number) but is a function and, as x solving exponential equations. as an inverse function to raising a number to a power where the exponent is the an inverse function to raising a number a domain and range, its input values produce unique output values, and it has it and values, output unique produce values input its range, and domain a an inverse function f Concepts and ideas that build towards functions leads to understanding the logarithmic function as a key tool for functions leads to understanding the logarithmic function as a functions as well as their graphs provide insight that can be used as the basis be used as the basis insight that can graphs provide as well as their functions of students’ share evidence we discuss and paper, In this for instruction. paper we focus on the importance of understanding the notion of function, function, of notion the understanding of importance the on focus we paper difficulties collected from various courses over time. We share concepts time. from various courses over difficulties collected only if x of logarithms. For example, students can understand why log example, because we know that 5 of a positive number x explore logarithms with students. explore logarithms of challenges in understanding logarithms as real numbers and logarithmic and logarithmic as real numbers logarithms in understanding of challenges output of the function. Formally, a logarithmic function with base a function a logarithmic of the function. Formally, output can also be composed with other functions, including its inverse relationship can also be composed with other functions, including its inverse mathematical notation, and properties used in relation to functions in order mathematical notation, and properties base) must be raised in order to produce the number x base) must be raised in order to bases by recognising that the inverse relationship inverse the that recognising by bases understanding these definitions, which involve many different mathematical concepts. In this these definitions, which involve to work with logarithms. these functions, and we present some ways that misconceptions related to and we present some ways that these functions, can listen for as they manifested to suggest what teachers these concepts are to the first equation). We also define a logarithm as a function—specifically We also define a logarithm to the first equation). rich understanding of logarithms relies on students’ understanding of both of rich understanding of logarithms relationship as a function. For example, understanding that understanding example, For function. a as relationship related to logarithms that could help students build an understanding of that could help students related to logarithms logarithm?” repeatedly. Typically it is defined as an exponent: The logarithm as an exponent: is defined it Typically repeatedly. logarithm?” with base 5 of 25 is 2 (that is, the equation log equation the is, (that 2 is 25 of 5 base with Developing an understanding of functions includes being able to classify a Developing an understanding of functions includes being able b x Teachers all over the globe undoubtedly hear the question, “What is a all over the globe undoubtedly hear the question, Teachers What is a logarithm? What is a logarithm? 14 Australian Senior Mathematics Journal vol. 27 no. 1 Kenney & Kastberg (mn) 4 couldberaisedtoproduce-16,believingthat the answershouldeitherbe Logarithms asinversefunctions . While interviewing twostudents, thefirstauthorfoundthatboth . Whileinterviewing convert This suggestsamisunderstanding oftheconceptinverserelationship 2 Many studentsunderstandlogarithmsassomething youprimarilydoor For instance,log would be an answer if they kept looking, but they were not able to find one. notation cancausegreatconfusionformanystudents(Tall &Razali,1993). the functionalrelationshipf to solveexponentialandlogarithmicequations(Weber, 2002a).Because understanding of exponents and inverse functions to build an understanding the same type of conversion by using guess and check to finda value towhich to propertiesofexponentials,canbewrittenaslog both a domain value for the logarithmic function and the product of by theirownuniquesetofpropertiesthatdistinguishthisfunctionfrom building an understanding of the unique characteristics of the logarithmic be fareasierandmoreaccuratethanmultiplyingthem).Properties be applied(forexample,insituationswhereaddingtwonumbersmight of connections betweenthelogarithmicandexponentialformsareneeded expressions, but also to recognise problem situations where logarithms can of logarithmicfunctions (Weber, 2000a;2000b). Yet iflogarithmsrepresent of exponentiation.Teachers hopethatstudentswill beabletousetheir could easilysolveproblemslikelog object and a process (Liang & Wood, 2005; Tall & Razali, 1993;Weber, 2002b). polynomial andotherfunctionsisasignificantcharacteristic.Forexample, is beingabletointerpretthelogarithmicnotationasrepresentingbothan interpret the notation and symbols involved. A critical component of this for functions oftheformlog families: Forf Student difficulties and exponentialfunctionscanbechallenging. a negativenumberorfraction(Kenney, 2006).Bothsuggestedthatthere an indicatorofthevalueusedinexponentiationprocess.Inalllogarithmic students havemoreexperiencewithalgebraicfunctionsandtheirproperties, students to understand and be able to use these properties to simplify 3 = The developmentoftheideathatlogarithmicfunctionsarecharacterized Understanding logarithmicfunctionsreliesinpartonbeingableto a (Weber, 2002b).Trying tomakesenseofthisdualroleplayed bythe f (x x 2 +2(mn)1andthensimplifyusingalgebraicproperties.However, + 7. However, whengiven log ) =log (x b (x ) =x a ), whenwereplacetheinputgetlog isusedasareferenttospecificlogarithmicfunctionand 2 +2x a +1,wecanreplacetheinputx (x (mn) isapplieddifferentlyfordifferentfunction ) =y , theinputvaluex 2 (x 4 (–16), both students tried to complete +7)=3byconvertingtheexpressionto >0canbethoughtofas b (m ) +log withmntoproduce b (mn) which,due b (n ). We want y factors

Links in learning logarithms Australian Senior Mathematics Journal vol. 27 no. 1 15 ]. (–1) inverse [labels the and this is the f -axis] and then it will have symmetry and the logarithmic function f inverse is going to go through (1,0) and it is going Figure 1. Nora’s drawing of a logarithmic function. Nora’s 1. Figure ]. = x = 0]. And that is going to go through (0,1) [draws exponential at [y function]. And f is going to look something like… It is going to have an asymptote is going to look something like… It is going to have an asymptote exponential function f to have an asymptote like [the y So which one of those is the log graph? Or is it all the log graph? It is all log, but this is just the f I don’t remember too much about the graph… Just the basic log graph remember too much about I don’t [draws y For Nora, the image created was “all log.” When asked to graph a For Nora, the image created was “all log.” When asked to Nora’s discussion of her drawing reveals that she can regenerate the image discussion of her drawing Nora’s Teachers often use graphs with logarithms to support the development of development the support to logarithms with graphs use often Teachers such as restrictions on the domain, may not develop. domain, may not on the such as restrictions a relationship between the function and its inverse. In Figure 1, we see that the function and its inverse. a relationship between (Kastberg, 2002). are inverse functions S: questions involving logarithms. For example, if asked to use this graph to find questions involving logarithms. For example, if asked to use this graph many textbooks. The image intends to convey the notion that the two curves The image intends to convey the many textbooks. only an action such as converting logarithmic to exponential form as in the form as in to exponential logarithmic action such as converting only an understandings, object, certain an inverse function above, rather than example understanding of the image would allow her to use it to check or explore understanding of the image would allow her to use it to check functions in the teacher to focus efforts on differentiating between the two the image. the teacher shared in class. log(–10) Nora might initially incorrectly assign a value. This would prompt log(–10) Nora might initially incorrectly assign a value. This would logarithm function, she always drew both an exponential and a logarithm. a and exponential an both drew always she function, logarithm Nora, a college algebra student, has created an image similar to those seen in student, has created an image Nora, a college algebra N: N: While her ability to generate the image is a good starting point, building an While her ability to generate the image is a good starting point, 16 Australian Senior Mathematics Journal vol. 27 no. 1 Kenney & Kastberg “cancelled out”(Kenney, with onestudent,Lynn, 2006).Afollow-upinterview “bereft of a succinct way to verbalize the operation performed ontheinput” “bereft ofasuccinctwaytoverbalizetheoperationperformed (p. 344) since no clear operation is signified in Logarithmic notation When aclassof59collegealgebrastudentswereaskedtoidentifywhether x I: I: L: L: L: sharedthefollowing: Lynn log (log role ofthesymbolsinvolvedinlogarithmicnotation, studentsfallbackon the twoexpressionswereequalorunequal,16studentsclaimedthat notation alsoallowsstudentstoworkwithfunctionsasobjectsorprocesses by transcendental functionssuchasf than amultiplicationsymbol.Itisalsosimilartotheirinteractionwithother unlike thechallengestudentsfacewhentheybegintointerprety that ln(x their understandingofpolynomialstoreasonabout logarithms. that studentoftendividedanequationlikeln(7x class discussions.However thestudents predominantly interpreted lnandlog expressions were equal, reasoning that log was irrelevant because it could be example, existing understandingsdrawnfrompolynomials.Forexample,considerthe of errorsmadebyAustralianstudentsonamathematics examinationandsaw expressions in order to act on the remaining terms. Given ln(x into playwithlogarithmicfunctionsbecausewehavespecialuniquenotation it wasavariableintheequation.Inabsenceof clearunderstandingofthe ignoring the“f for certainbases,namelylog(x following unequalexpressions:log further demonstratedhowstudentsmighteliminatethelognotationfrom Students begintobuildtheirunderstandingoffunctionnotationastheymake activities. Studentswerealso askedtoidentifythebaseofeachlogarithmin all sense of how to evaluate functions using the notation (Hurwitz 1999).For sense ofhowtoevaluatefunctionsusingthenotation(Hurwitz side. Hurwitz (1999)findsthatlogarithmicnotation,however,side. Hurwitz leavesstudents ismultipliedby2signifiedthejuxtapositionofx This experienceissimilartoafindingfrom Yen (1999)whoanalysedtypes In the same algebra survey (Kenney,In thesamealgebrasurvey 2006),35outof59studentsanswered Students struggletointerpretlogarithmicnotationandoftenreachfortheir I guesstheyjustdisappear. Where dotheygo? They cancelout. What happenstotheln-sagain? The ln-sare…theycancelouttogiveyoux would be0soyou’djusthavetheplus3,well,–3. e ) andcommonlog(log ) was equal to log(x f (x ) =2x (x )” symbolsontheleftanddealingonlywithrighthand +3tellsthelearnertodoubleaninputandaddthree.This ) although these special notations for the natural ) forlog 10 ) weredistinguishedinclass discussionsand (x 3 (x ) =sin(x ) +log 10 (x ) andln(x 3 ). Additionalcomplicationscome (x +1),andlog –3)=ln(2x minusx f (x ) forlog ) = log +3,sox 5 andthe2rather ) bylnasthough 2 (x (x e (x ) +log ). This is not ). ) – ln(x =2x minusx 5 (x , where +1). + 3), Links in learning logarithms Australian Senior Mathematics Journal vol. 27 no. 1 17

+ y a + 5log (2) was not x and asked: 3 a 4 z 5 (1) because the b 6xy a 6 + log a (1) was different from log a ) can appear reasonable when one’s thinking ) can appear reasonable when one’s (N b ) + log (M b (2) because the bases were different. Yet, 26 students used this 26 students used this Yet, (2) because the bases were different. 4 ) = log + N . After the teacher finished, Jamie asked: “So if it says write as a sum, you asked: “So if it says write as . After the teacher finished, Jamie z a (M b Properties are powerful tools students use to simplify expressions and same reasoning to say that log support students to understand the logarithmic function. Misconceptions support students to understand the logarithmic function. Misconceptions solve equations, however overgeneralisation of polynomial ideas applied solve equations, however overgeneralisation algebra survey, 57 out of 59 students correctly answered that log 57 algebra survey, as interchangeable symbols. Teachers need to help students recognize that that recognize students help need to Teachers symbols. interchangeable as about numbers and polynomials has been based on the distributive property. been based on the distributive property. about numbers and polynomials has different ideas (Stacey & MacGregor, 1997). In addition, gathering student addition, gathering 1997). In MacGregor, ideas (Stacey & different time for the development of meaning of the symbols allows insights about the for them. personal meaning ideas to make sense of the new. So, for example, a relationship such as So, for example, ideas to make sense of the new. don’t work it out? Just go that one step? I worked it out and found a number.” one step? I worked it out and found a number.” work it out? Just go that don’t considering the meaning of the new notations they are using. expressions and support the solving of equations. Yet without connections without connections Yet support the solving of equations. expressions and prior knowledge (Liang & from instruction and from overgeneralizations different results. In the college bases yield different using same quantity of the obtain a numeric answer (Kastberg, 2002). equal to log mathematics has a specialized language and uses conventions to represent to conventions uses and language specialized a has mathematics by doing the problem on the board. Her answer was log by doing the problem on the board. based on prior knowledge are a part of developing understanding. based on prior knowledge are a part of developing understanding. between ideas and development of these properties, there is room for properties, there is room for of these development between ideas and bases were different (Kenney, 2006). bases were different (Kenney, the properties were used to “convert” the expression into an equation and the properties were used to “convert” Building deeper understanding to notation coupled with new properties can cause confusion. During class, to notation coupled with new properties log For Jamie the idea of an expression and an equation were conflated and so For Jamie the idea of an expression But now learners must differentiate that knowledge and reason in new ways by But now learners must differentiate that knowledge and reason in Jamie was confused about expanding the expression log Jamie was confused about expanding When mathematics learners of any age develop new ideas, they use old When mathematics learners of any age develop new ideas, they Wood, 2005). One such example is the overgeneralisation of the idea that logs 2005). One such Wood, These examples illustrate common misconceptions that emerge as teachers These examples illustrate common misconceptions that emerge The unique properties of logarithms are powerfulare logarithms of properties unique The to transform ability their in 4log Unique properties of logarithms Unique properties “Are you supposed to work it out?” The teacher responded to Jamie’s question The teacher responded to Jamie’s “Are you supposed to work it out?” 18 Australian Senior Mathematics Journal vol. 27 no. 1 Kenney & Kastberg Attending todiscourse Building connectionstopriorexperiences This isasimpleexercise,butitbuildsonstudents’existingunderstandingof When teachinglogarithms,itisimportanttopay attentionbothtothewords It isimportantforstudentstorecognise the proceduressignifiedby 2002b). Whenstudentsarefirstintroducedtotheideaofavariable,weoften Linking thesquarerootnotationwithlnandlogcouldimprovestudents’ hear teachers make connections to everyday objects (e.g., “pretend I have hear teachers make connections to everyday have restrictionsontheinputvalue’s domain,andhaveuniquenotationthat however, includesviewinglogarithmsasobjectsthatcanbedecomposedand how tonotatethreeobjectsanddevelopsanunderstandingoflog we useandtheideasthatstudentsshareabout theirunderstanding.For like objects.Asimilarexercisecouldbeeffectivewithlogarithms.Consider root button on a calculator to perform theprocesssuggestedbysymbol. root buttononacalculatortoperform recomposed intonewobjects,followingthepropertiesoflogarithms,while them. they cannotberemovedfromequationswithoutsomeprocessactingupon understanding ofthedualrolethatthesesymbolsplayandrecognition to therequesttakelogofbothsidesanequation),andusedsquare thinking. Showingstudents howtoconstructthegraphoflog told thatthey can solveanequationlike 3 this mayleadtoincorrect conceptionslikeNora’s. Similarly, studentsareoften builds meaningforthenewideas. becomes areasontoincorrectlyremovethelog term fromalogequation.It one apple,andtwomoreapples,howmanyappleswouldIhave?”)tohelp exponential graphasin Figure 1isimportant,butwithoutfurtherdiscussion example, weoftenhearteachersrefertothefact that log object itself.Askingstudentstoevaluatebothanswersforx itself resultsintheinputvalue).Studentshave‘taken’squareroot(similar indicate theprocesstotake(e.g.,findnumberthatwhenmultipliedby parallels orcomparingtostudentsexistingunderstandingandnewideas ideas aboutwhatitmeansto‘cancel’andwhen canandcannotbeused. is importanttoquestionstudentswhentheyuse this term,tochallengetheir inverse relationships‘cancel’eachotherout.For students,thetermcancel function notation. Conceptual understanding of logarithmic functions, asking studentstosimplifyx also anticipatingtheexponentiationprocess needed tosolveforx away. Thismayalsoallowstudentstolearninterpretthenotation.Drawing an understandingoftheroleplayedbylog students buildtheideaofan‘x square root.Likelogarithms,rootsrepresentaninverserelationship, Teachers canalsoconnecttostudents’experienceswithfunctionssuchas Teachers’ useofrepresentations cansometimesleadtolimitationsinstudent +x ’ asanobjectthatcancombinedwithother +x , andthenlog x =4 2 andthatitcannotbe‘cancelled’ x –2 bytakingeitherthelogor ln 2 (x ) +log b (b =2canalsobuild x ) =x 2 b (x (x becausethe ) +log ) fromthe 2 (x (Weber, ) asan 2 (x ). Links in learning logarithms Australian Senior Mathematics Journal vol. 27 no. 1 19 + 1)? (x 3 ) and log (x 3 ) without starting with the exponential ) without starting with the exponential (x 2 ) as the exact same functions. ) as the exact same . N b ) and ln(x + log (x M 10 b mean in the expressions log mean in the expressions 3 ) ≠ log + N (M b graph? number. log ln(3). Describe what you think your calculator is doing when you use it to find think your calculator is doing Describe what you Follow-up questions during class discussions and open-ended assessment and open-ended assessment discussions during class Follow-up questions Produce an argument that could convince a friend that Produce an argument that could How could we plot the graph of log Give an example of an exponential equation whose solution is a negative Give an example of an exponential What does log • • • • • secondaryan understanding of logarithms need to build from mathematics significantly in calculus), science, and engineering. It is therefore important significantly in calculus), science, students who saw log students who saw as real numbers toward an understanding of logarithms as the range of a as real numbers toward an understanding of logarithms as the about logarithms and logarithmic functions. Consider how the students whose students Consider how the functions. logarithmic and about logarithms Summary for students to understand logarithms as real numbers as well as the for students to understand logarithms informal questioning can result in more flexible student understanding of informal questioning can result in more flexible student understanding items can help a teacher know what understandings students have developed teacher know what understandings items can help a ideas by relating to prior knowledge. Students moving from Year 10 to senior Year moving from Students ideas by relating to prior knowledge. characteristics of logarithmic functions. As teachers, we want students to characteristics of logarithmic functions. of both sides. These two choices are usually offered because they are available they are available offered because choices are usually These two of both sides. of the base used. the same regardless the answer is and because on a calculator be able to develop a differentiated understanding of logarithmic functions logarithmic of understanding differentiated a develop able to be understandings of these new objects using processes that are more familiar, to familiar, understandings of these new objects using processes that are more through this, teachers are encouraged to explore results of students’ efforts this unique and powerful function. thinking we have described might answer some of the following questions: described might answer some of thinking we have rather than a set of discrete skills, and to be able to regenerate and self-check rather than a set of discrete skills, logarithmic function. It is important that students take their time to build to time their take students that important is It function. logarithmic achieve To learn to notate their reasoning rather than copying images provided. Logarithms continue to play an important role in mathematics (most Logarithms continue to play However, statements like these could lead to the conception portrayed by the conception portrayed lead to the like these could statements However, References

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