Vocabulary to Review

Objective 10.1A

Vocabulary to Review irrational number [1.1A]

New Vocabulary exponential function natural exponential function

New Symbols e

Discuss the Concepts 1. How does an exponential function differ from a polynomial function? 2. Why are the conditions b > 0, b ≠ 1 given for f(x) = bx? 3. Is it possible to express e as the quotient of two integers? Why or why not?

Concept Check 1. For the functions g( x) = 3x and h( x) = 4x , which function has the greater values when a. x > 0? h(x) b. x < 0? g(x)

2. Between what two consecutive integers are some values of the function f( x) = x3 greater than the corresponding values of the function P( x) = 3x ? 2 and 3

Optional Student Activity Suppose a culture of bacteria starts with 30 bacteria and the population grows according to the function f( x) = (30) 2x , where x is the time in hours. a. What is the rate of change between x = 1 and x = 2 ? 60 bacteria per hour b. What is the rate of change between x = 2 and x = 3 ? 120 bacteria per hour c. What is the rate of change between x = 3 and x = 4 ? 240 bacteria per hour d. Express the rate of change between x= n and x= n +1 as an exponential function. f(n) = (60)2n-1

Objective 10.1B

Vocabulary to Review vertical-line test [9.2A] horizontal-line test [9.4A] 1–1 function [9.4A]

Discuss the Concepts Determine whether the statement is always true, sometimes true, or never true. 1. The domain of an exponential function f(x) = bx, b > 0, b ≠ 1, is the set of positive numbers. Never true 2. An exponential function f(x) = bx, b > 0, b ≠ 1, is a 1–1 function. Always true 3. The graph of an exponential function f(x) = bx, b > 0, b ≠ 1, passes through the point (0, 0). Never true 4. For the function f(x) = bx, b > 0, b ≠ 1, the base b is a positive integer. Sometimes true 5. An exponential function f(x) = bx, b > 0, b ≠ 1, has two x-intercepts. Never true

Concept Check 1. Graph f(x) = 20.5x -1 – 4. For what values in the domain are the corresponding values in the range less than 0? None 2. Graph f(x) = ex – 1. What are the x- and y-intercepts of the graph of the function? x-intercept: (0, 0); y-intercept: (0, 0) 3. What is the y-intercept of the graph of f(x) = e-2x – 4? (0, -3)

Optional Student Activity For each of the following functions, determine whether the values of y increase or decrease as x increases. Then try to state a rule that summarizes your findings. a. f( x) = 0 . 5x b. g( x) = 4x

x 骣3 c. h( x ) = 琪桫4 d. F( x) = 2x

x 骣1 e. G( x ) = 琪桫3 f. H( x) = 1 . 5x

Values of y increase as x increases for functions b, d, and f. Values of y decrease as x increases for functions a, c, and e. Values of y increase as x increases when the base of the exponential function is greater than 1. Values of y decrease when the base is between 0 and 1.

Answers to Writing Exercises 1. An exponential function with base b is defined by f(x) = bx, b > 0, b ≠ 1, and x is any real number. 2. The natural exponential function is the function defined by f(x) = ex, where e is an irrational number approximately equal to 2.71828183.

Answers to Writing Exercises 41. b. The point (2, 27.7) means that after 2 s, the object is falling at a speed of 27.7 ft/s.

Objective 10.2A

Vocabulary to Review 1–1 function [9.4A] inverse function [9.4B]

New Vocabulary logarithm common logarithms natural logarithm

New Properties 1–1 Property of Exponential Functions Discuss the Concepts Explain why each of the following is true. 1. ln ex = x 2. ln e = 1 3. eln x = x

Optional Student Activity Determine the domain of the function. Recall that the logarithm of a nonpositive number is not defined.

1. f(x) 5 log3(x – 4) {x | x > 4}

2. f(x) 5 log2(x + 2) {x | x > -2} 3. f(x) 5 ln(x2 – 4) {x | x < -2 or x > 2} 4. f(x) 5 ln(x2 + 4) {x | x  real numbers}

5. f(x) 5 log2 x 1 log2(x – 1) {x | x > 1} x x + 2 6. f(x) 5 log4 {x | x < -2 or x > 0}

Objective 10.2B

New Properties The Logarithm Property of the Product of Two Numbers The Logarithm Property of the Quotient of Two Numbers The Logarithm Property of the Power of a Number Inverse Property of Logarithms 1–1 Property of Logarithms

Discuss the Concepts Determine whether the statement is always true, sometimes true, or never true.

x logb x 1. logb = Never true ylogb y

log x x 2. = Sometimes true log y y

3. log(x+ y ) = log x + log y Never true

1 4. logx= log x Always true b2 b

Concept Check Use the Properties of Logarithms to solve for x.

1. log8 x = 3 log8 2 8

2. log5 x = 2 log5 3 9

3. log4 x = log4 2 + log4 3 6 4. log3 x = log3 4 + log3 7 28

5. log6 x = 3 log6 2 – log6 4 2

6. log9 x = 5 log9 2 – log9 8 4 1 7. log x = log 27 3 3 3 8. log2 x = log2 4 8 2

Concept Check 1/3 1. Evaluate: log8 2 – log6 216 + log3 81 – log5(625) 0 1

7 2. Evaluate: log9 9 – log4 64 + log2 2 – log3 1 3

Optional Student Activity x 1. Use the Properties of Logarithms to show that loga a = x, a > 0 . x loga a = x loga a = x(1) = x

2. Use the Properties of Logarithms to show that blogb x = x, b > 0 , x > 0 . y logb x The logarithmic form of b = x is logbx = y. Use this fact to write b = x in exponential form: logb x = logb x, a true equation. (Note that we are simply substituting logb x for y.)

Objective 10.2C

Vocabulary to Review common logarithm [10.2A] natural logarithm [10.2A]

New Formulas Change-of-Base Formula

logb N loga N = logba

Discuss the Concepts Explain how to use the Change-of-Base Formula to evaluate log5 12.

Optional Student Activity

1. Given f(x) = 3 log6(2x – 1), determine f (7) to the nearest hundredth. 4.29

2. Given S(t) = 8 log5(6t + 2), determine S(2) to the nearest hundredth. 13.12

3. Given P(v) = -3 log6(4 – 2v), determine P(- 4) to the nearest hundredth. -4.16

4. Given G(x) = -5 log7(2x + 19), determine G(- 3) to the nearest hundredth. -6.59 Answers to Writing Exercises 1. a. A common logarithm is a logarithm with base 10. b. log 4z 2. a. A natural logarithm is a logarithm with base e. b. ln 3x

Answers to Writing Exercises

47. Answers will vary. For example, the log of a product is equal to the sum of the logs: logb(xy) = logb x + logb y. 48. Answers will vary. For example, the log of a quotient is equal to the difference of the logs: x y logb = logb x – logb y.

Objective 10.3A

Vocabulary to Review equivalent exponential and logarithmic equations [10.2A]

Discuss the Concepts 1. The inverse of an exponential function is a logarithmic function. Why?

2. Is it true that if x and y are positive real numbers x < y, and b > 0, then logb x , logb y? Yes

3. The Power Property of Logarithms tells us that ln x2 = 2 ln x. Graph f(x) = ln x2 and g(x) = 2 ln x. Are the graphs the same? Explain. No. If x < 0, ln x2 is a real number but 2 ln x is not a real number.

x -1 4. Does the graph of the exponential function f(x) = b intersect the graph of its inverse f (x) = logb x when b > 1? No

x -1 5. Does the graph of the exponential function f(x) = b intersect the graph of its inverse f (x) = logb x when 0 < b < 1? Yes Concept Check 1. Graph f(x) = 10 log(x – 2). Find the zero of the function. 3 1 2. What value in the domain of f(x) = 3 log2 x corresponds to the range value of -6? 4

3. What value in the domain of f(x) = log3(2 – x) corresponds to the range value of 1? -1

4. What value in the domain of f(x) = -log2(x – 1) corresponds to the range value of -2? 5 5. Graph f(x) = 5 log(x – 1). What are the x- and y-intercepts of the graph of this function? x-intercept: (2, 0); no y-intercept

Optional Student Activity According to the U.S. Environmental Protection Agency, the amount of garbage generated per person has been increasing over the last few decades. The table below shows the per capita garbage, in pounds per day, generated in the United States for selected years. Pounds Per Year Day 1960 2.66 1970 3.27 1980 3.61 1990 4.00 Draw a scatter diagram for these data. Would the equation that best fits the points be the equation of a linear function, an exponential function, or a logarithmic function? Linear

Answers to Writing Exercises 1. Yes. Answers will vary. For example, the function passes both the vertical-line test and the horizontal-line test. 2. Descriptions will vary. For example: The domain is the set of positive real numbers; the range is the set of real numbers. The graph never touches the y-axis. The x-intercept is (1, 0). 3. They are the same graph. 4. They are mirror images of each other with respect to the line y = x.

Answers to Writing Exercises 23. b. The point (25.1, 2) means that a star that is 25.1 parsecs from Earth has a distance modulus of 2. 24. b. The point (4, 49) means that after 4 months, the typist’s proficiency has dropped to 49 words per minute.

Objective 10.4A

New Vocabulary exponential equation

Properties to Review 1–1 Property of Exponential Functions [10.2A] Discuss the Concepts 1. Explain how to solve the equation 7x+1= 7 5 .

2. What is the solution of the equation 2ex = 16 ? Explain how you determined the solution. ln 8. Divide both sides of the equation by 2; then write the equation in logarithmic form. 3. What does the following property allow us to do? If x = y, x > 0, y > 0, then log x = log y. Take the common logarithm of each side of an equation

Concept Check 1. If 3x = 5, find the value of 32x+ 3 . 675

2. Given 2x= 8 y+1 and 9y= 3 x-9 , find the value of x+ y. 27

3. Find the x-coordinate of the intersection of the graphs of y = 42x and y= 8x+1 . 3

Optional Student Activity 1. When all eight positive integer factors of 30 are multiplied together, the product is 30k . Find k. 4

2. Solve: 4x/ 3 = 2 1.5 3. Solve for x. Round to the nearest ten-thousandth.1. 2x/ 2- 1 = 1 . 4 5.6910

4. Solve for x: 5x = 599+ 5 99 + 599+ 5 99 + 5 99 100

Objective 10.4B

Properties to Review 1–1 Property of Logarithms [10.2B]

Discuss the Concepts

1. Explain how to solve the equation 2 = log3 x.

2. Can the 1–1 Property of Logarithms be used to solve the equation log3 x = log2 8? Why or why not? No. The bases are not the same. 3. If log(x + 2) + log(x – 2) = 6, then what expression is equal to 106? x2 - 4 4. Why is it possible for a logarithmic equation to have an extraneous solution?

Concept Check Determine whether the statement is true or false given that x, y, and b are positive real numbers and b?1 . Explain your reasoning. 1. log x + log(x + 2) = log(2x + 2) False 2x - 2 2. log(2x – 2) – log x = 4 is equivalent to log = 4. True x

3. log(2x) + log 4 = 6 is equivalent to log(8x) = 6. True

Optional Student Activity Solve the system of equations. 1. log(x + y) = 3 x = y + 4 (502, 498) 2. log(x + y) = 3 x – y = 20 (510, 490) 3. 83x= 4 2 y x- y = 5 (-4, -9) 4. 93x= 81 3 y x+ y = 3 (2, 1)

Answers to Writing Exercises 1. An exponential equation is one in which a variable occurs in an exponent. 2. a. The 1–1 Property of Exponential Functions states that for b > 0, b ≠ 1, if bx= b y , then x= y. b. This property is used to solve exponential equations in which each side of the equation can be expressed in terms of the same base. 27. A logarithmic equation is an equation in which one or more of the terms is a logarithmic expression.

28. The 1–1 Property of Logarithms states that for b > 0, b ≠ 1, if logb x = logb y, then x= y.

Answers to Writing Exercises 55. The error is in the second step because the value of log 0.5 is less than zero. Multiplying each side of the inequality by this negative quantity changes the direction of the inequality symbol.

Objective 10.5A

New Vocabulary exponential growth equation exponential decay equation compound interest compound interest formula

New Formulas n compound interest formula, A= P(1 + i) pH formula, pH = –log(H+) I Richter scale magniture, M = log I0

Discuss the Concepts 1. State whether each of the following equations is an exponential growth equation, an exponential decay equation, or neither. Explain your reasoning. a. f( x) = 6x

x 骣2 b. g( x ) = 琪桫3 c. F( x) = 3 x4 d. f( x) = ex

x e. y = 250( 1.008)

r 骣1 f. t( r )= 64琪桫2 a, d, and e are exponential growth equations. b and f are exponential decay equations. c is neither. 2. The intensity I of an x-ray after it has passed through a material that is x centimeters thick is given by I = -kx I0e , where I0 is the initial intensity of the x-ray and k is a number that depends on the material. Radiologists (physicians who specialize in the use of radioactive substances to diagnose and treat disease) wear lead shields when giving patients x-rays. The constant k for lead is 43. Use the given equation to explain why a lead shield of the same thickness as a copper shield (k = 3.2) offers better protection to the radiologist than the copper shield.

Concept Check The atmospheric pressure P decreases exponentially with height h above sea level. The equation relating the pressure P, in pounds per square inch, and height h, in feet, is P = 14.7e-0.00004h. Find the height of Mt. Everest if the atmospheric pressure at the top is 4.6 lb/in2.. Round to the nearest foot. 29,045 ft

Optional Student Activity 1. An annuity is a fixed amount of money that is either paid or received over equal intervals of time. A retirement plan into which a certain amount is deposited each month is an example of an annuity; equal deposits are made over equal intervals of time (monthly). The equation that relates the amount of money 轾(1+i)x - 1 available for retirement to the monthly deposit is V= P 犏 , where i is the interest rate per i 臌犏 month, x is the number of months deposits are made, P is the payment, and V is the value (called the future value) of the retirement fund after x payments. Suppose $100 is deposited each month into an account that earns interest at the rate of 0.5% per month (6% per year). For how many years must the investor make deposits in order to have a retirement account worth $20,000? Round to the nearest whole number. 12 years 2. One scientific study suggests that the carrying capacity of Earth is approximately 10 billion people. Find out what is meant by “carrying capacity.” Research the current world population and project when Earth’s population will reach 10 billion, assuming population growth rates of 1%, 2%, 3%, 4%, and 5%. Research the current rate of world population growth and use that number to determine when, according to the model, the world population will reach 10 billion.

Answers to Focus on Problem Solving: Proof by Contradiction a. The sum of the 10 digits is 45. Let x = the sum of the tens digits, and S = the sum of the remaining units digits. Then S = 45 – x. b. 10x+( 45 - x) = 100 55 c. x = 9 55 d. x is the sum of integers, so x = is not possible. Our assumption that it is possible to use each of the 9 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 exactly once in such a way that the sum is 100 is not valid.

Answers to Projects and Group Activities: Solving Exponential and Logarithmic Equations Using a Graphing Calculator 1. -1.86, 3.44 2. -1.25 3. -1.16 4. -1.24, 2.42 5. 2.42 6. 0.19 7. -1.51, 2.10 8. 1.00

Answers to Projects and Group Activities: Credit Reports and FICO® Scores 1. The equation is y = 918.22 – 137.49 ln x, where values are rounded to the nearest hundredth. 2. 64% 3. 612 4. 549 5. 700 6. Three major credit reporting agencies are Equifax, Experian, and Trans Union.