Figure 1 electron micrograph of E. Coli bacteria (credit: “mattosaurus,” Wikimedia Commons)

Figure 1 electron micrograph of E. Coli bacteria (credit: “mattosaurus,” Wikimedia Commons)

6.1 exponential Functions 6.5 logarithmic Properties 6.2 graphs of exponential Functions 6.6 exponential and logarithmic equations 6.3 logarithmic Functions 6.7 exponential and logarithmic models 6.4 graphs of logarithmic Functions 6.8 Fitting exponential models to data

Introduction Focus in on a square centimeter of your skin. Look closer. Closer still. If you could look closely enough, you would see hundreds of thousands of microscopic organisms. They are bacteria, and they are not only on your skin, but in your mouth, nose, and even your intestines. In fact, the bacterial cells in your body at any given moment outnumber your own cells. But that is no reason to feel bad about yourself. While some bacteria can cause illness, many are healthy and even essential to the body.

Bacteria commonly reproduce through a process called binary fission, during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours, as opposed to days or years.[16]

For simplicity’s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. Table 1 shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! And if we were to extrapolate the table to twenty-four hours, we would have over 16 million!

Hour 0 1 2 3 4 5 6 7 8 9 10 Bacteria 1 2 4 8 16 32 64 128 256 512 1024

Table 1

In this chapter, we will explore exponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions, which are closely related to exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and interpreting data. 16 Todar, PhD, Kenneth. Todar’s Online Textbook of Bacteriology. http://textbookofbacteriology.net/growth_3.html.

Exponential and Logarithmic Functions

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476 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

6.1 SeCTIOn exeRCISeS

veRbAl 1. Explain why the values of an increasing exponential

function will eventually overtake the values of an increasing linear function.

2. Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.

3. The Oxford Dictionary defines the word nominal as a value that is “stated or expressed but not necessarily corresponding exactly to the real value.”[18] Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.

AlgebRAIC For the following exercises, identify whether the statement represents an exponential function. Explain.

4. The average annual population increase of a pack of wolves is 25.

5. A population of bacteria decreases by a factor of 1 __ 8  every 24 hours.

6. The value of a coin collection has increased by 3.25% annually over the last 20 years.

7. For each training session, a personal trainer charges his clients $5 less than the previous training session.

8. The height of a projectile at time t is represented by the function h(t) = −4.9t 2 + 18t + 40.

For the following exercises, consider this scenario: For each year t, the population of a forest of trees is represented by the function A(t) = 115(1.025)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 82(1.029)t. (Round answers to the nearest whole number.)

9. Which forest’s population is growing at a faster rate? 10. Which forest had a greater number of trees initially? By how many?

11. Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 20 years? By how many?

12. Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 100 years? By how many?

13. Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.

14. y = 300(1 − t)5 15. y = 220(1.06)x

16. y = 16.5(1.025) 1 _ x 17. y = 11,701(0.97)t

For the following exercises, find the formula for an exponential function that passes through the two points given.

18. (0, 6) and (3, 750) 19. (0, 2000) and (2, 20) 20.  −1, 3 _ 2  and (3, 24) 21. (−2, 6) and (3, 1) 22. (3, 1) and (5, 4)

18. Oxford Dictionary. http://oxforddictionaries.com/us/definition/american_english/nomina.

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SECTION 6.1 sectioN exercises 477

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

23. x 1 2 3 4 f (x) 70 40 10 −20

24. x 1 2 3 4 h(x) 70 49 34.3 24.01

25. x 1 2 3 4 m (x) 80 61 42.9 25.61

26. x 1 2 3 4 f (x) 10 20 40 80

27. x 1 2 3 4 g (x) −3.25 2 7.25 12.5

For the following exercises, use the compound interest formula, A(t) = P  1 + r _ n  nt.

28. After a certain number of years, the value of an investment account is represented by the equation 10, 250  1 + 0.04 ____ 12 

120. What is the value of the account?

29. What was the initial deposit made to the account in the previous exercise?

30. How many years had the account from the previous exercise been accumulating interest?

31. An account is opened with an initial deposit of $6,500 and earns 3.6% interest compounded semi-annually. What will the account be worth in 20 years?

32. How much more would the account in the previous exercise have been worth if the interest were compounding weekly?

33. Solve the compound interest formula for the principal, P.

34. Use the formula found in Exercise #31 to calculate the initial deposit of an account that is worth $14,472.74 after earning 5.5% interest compounded monthly for 5 years. (Round to the nearest dollar.)

35. How much more would the account in Exercises #31 and #34 be worth if it were earning interest for 5 more years?

36. Use properties of rational exponents to solve the compound interest formula for the interest rate, r.

37. Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $9,000 and was worth $13,373.53 after 10 years.

38. Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of $5,500, and was worth $38,455 after 30 years.

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

39. y = 3742(e)0.75t 40. y = 150 (e) 3.25 _ t 41. y = 2.25(e)−2t

42. Suppose an investment account is opened with an initial deposit of $12,000 earning 7.2% interest compounded continuously. How much will the account be worth after 30 years?

43. How much less would the account from Exercise 42 be worth after 30 years if it were compounded monthly instead?

nUmeRIC For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.

44. f (x) = 2(5)x, for f (−3) 45. f (x) = −42x + 3, for f (−1) 46. f (x) = e x, for f (3)

47. f (x) = −2e x − 1, for f (−1) 48. f (x) = 2.7(4)−x + 1 + 1.5, for f (−2) 49. f (x) = 1.2e2x − 0.3, for f (3)

50. f (x) = − 3 _ 2 (3) −x + 3 _ 2 , for f (2)

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478 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

TeChnOlOgy For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.

51. (0, 3) and (3, 375) 52. (3, 222.62) and (10, 77.456) 53. (20, 29.495) and (150, 730.89)

54. (5, 2.909) and (13, 0.005) 55. (11,310.035) and (25,356.3652)

exTenSIOnS 56. The annual percentage yield (APY) of an investment

account is a representation of the actual interest rate earned on a compounding account. It is based on a compounding period of one year. Show that the APY of an account that compounds monthly can be found with the formula APY =  1 + r __ 12 

12 − 1.

57. Repeat the previous exercise to find the formula for the APY of an account that compounds daily. Use the results from this and the previous exercise to develop a function I(n) for the APY of any account that compounds n times per year.

58. Recall that an exponential function is any equation written in the form f (x) = a . b x such that a and b are positive numbers and b ≠ 1. Any positive number b can be written as b = en for some value of n. Use this fact to rewrite the formula for an exponential function that uses the number e as a base.

59. In an exponential decay function, the base of the exponent is a value between 0 and 1. Thus, for some number b > 1, the exponential decay function can be written as f (x) = a .  1 _ b 

x . Use this formula, along

with the fact that b = e n, to show that an exponential decay function takes the form f (x) = a(e) −nx for some positive number n.

60. The formula for the amount A in an investment account with a nominal interest rate r at any time t is given by A(t) = a(e)rt, where a is the amount of principal initially deposited into an account that compounds continuously. Prove that the percentage of interest earned to principal at any time t can be calculated with the formula I(t) = e rt − 1.

ReAl-WORld APPlICATIOnS 61. The fox population in a certain region has an annual

growth rate of 9% per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year 2020?

62. A scientist begins with 100 milligrams of a radioactive substance that decays exponentially. After 35 hours, 50 mg of the substance remains. How many milligrams will remain after 54 hours?

63. In the year 1985, a house was valued at $110,000. By the year 2005, the value had appreciated to $145,000. What was the annual growth rate between 1985 and 2005? Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010?

64. A car was valued at $38,000 in the year 2007. By 2013, the value had depreciated to $11,000 If the car’s value continues to drop by the same percentage, what will it be worth by 2017?

65. Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?

66. Kyoko has $10,000 that she wants to invest. Her bank has several investment accounts to choose from, all compounding daily. Her goal is to have $15,000 by the time she finishes graduate school in 6 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal? (Hint : solve the compound interest formula for the interest rate.)

67. Alyssa opened a retirement account with 7.25% APR in the year 2000. Her initial deposit was $13,500. How much will the account be worth in 2025 if interest compounds monthly? How much more would she make if interest compounded continuously?

68. An investment account with an annual interest rate of 7% was opened with an initial deposit of $4,000 Compare the values of the account after 9 years when the interest is compounded annually, quarterly, monthly, and continuously.

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488 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

6.2 SeCTIOn exeRCISeS

veRbAl

1. What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

2. What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

AlgebRAIC

3. The graph of f (x) = 3x is reflected about the y-axis and stretched vertically by a factor of 4. What is the equation of the new function, g(x)? State its y-intercept, domain, and range.

4. The graph of f (x) =  1 _ 2  −x

is reflected about the y-axis and compressed vertically by a factor of 1 _ 5 . What is the equation of the new function, g(x)? State its y-intercept, domain, and range.

5. The graph of f (x) = 10x is reflected about the x-axis and shifted upward 7 units. What is the equation of the new function, g(x)? State its y-intercept, domain, and range.

6. The graph of f (x) = (1.68)x is shifted right 3 units, stretched vertically by a factor of 2, reflected about the x-axis, and then shifted downward 3 units. What is the equation of the new function, g(x)? State its y-intercept (to the nearest thousandth), domain, and range.

7. The graph of f (x) = − 1 _ 2  1 _ 4 

x − 2 + 4 is shifted

downward 4 units, and then shifted left 2 units, stretched vertically by a factor of 4, and reflected about the x-axis. What is the equation of the new function, g(x)? State its y-intercept, domain, and range.

gRAPhICAl

For the following exercises, graph the function and its reflection about the y-axis on the same axes, and give the y-intercept.

8. f (x) = 3  1 _ 2  x

9. g(x) = −2(0.25)x 10. h(x) = 6(1.75)−x

For the following exercises, graph each set of functions on the same axes.

11. f (x) = 3  1 _ 4  x , g(x) = 3(2)x, and h(x) = 3(4)x 12. f (x) = 1 _ 4 (3)

x, g(x) = 2(3)x, and h(x) = 4(3)x

For the following exercises, match each function with one of the graphs in Figure 12.

A

B C D E

F

Figure 12

13. f (x) = 2(0.69)x 14. f (x) = 2(1.28)x 15. f (x) = 2(0.81)x

16. f (x) = 4(1.28)x 17. f (x) = 2(1.59)x 18. f (x) = 4(0.69)x

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SECTION 6.2 sectioN exercises 489

For the following exercises, use the graphs shown in Figure 13. All have the form f (x) = ab x. y

x

A

B C D

E

F

Figure 13

19. Which graph has the largest value for b? 20. Which graph has the smallest value for b? 21. Which graph has the largest value for a? 22. Which graph has the smallest value for a?

For the following exercises, graph the function and its reflection about the x-axis on the same axes.

23. f (x) = 1 _ 2 (4) x 24. f (x) = 3(0.75)x − 1 25. f (x) = −4(2)x + 2

For the following exercises, graph the transformation of f (x) = 2x. Give the horizontal asymptote, the domain, and the range.

26. f (x) = 2−x 27. h(x) = 2x + 3 28. f (x) = 2x − 2

For the following exercises, describe the end behavior of the graphs of the functions.

29. f (x) = −5(4)x − 1 30. f (x) = 3  1 _ 2  x − 2 31. f (x) = 3(4)−x + 2

For the following exercises, start with the graph of f (x) = 4x. Then write a function that results from the given transformation.

32. Shift f (x) 4 units upward 33. Shift f (x) 3 units downward 34. Shift f (x) 2 units left 35. Shift f (x) 5 units right 36. Reflect f (x) about the x-axis 37. Reflect f (x) about the y-axis

For the following exercises, each graph is a transformation of y = 2x. Write an equation describing the transformation.

38.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

210 4

4

5

5

39.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

210 4

4

5

5

40.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

210 4

4

5

5

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490 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

For the following exercises, find an exponential equation for the graph.

41.

4

x

y

–1–2–2

–4

–3

–6

–4

–8

–5

–10

2

3

6

210 4

8

5

10

42.

2

x

y

–1–1–2

–2

–3

–3

–4

–4

–5

–5

1

3

3

210 4

4

5

5

nUmeRIC

For the following exercises, evaluate the exponential functions for the indicated value of x.

43. g (x) = 1 _ 3 (7) x − 2 for g(6). 44. f (x) = 4(2)x − 1 − 2 for f (5). 45. h(x) = − 1 _ 2 

1 _ 2  x + 6 for h(−7).

TeChnOlOgy

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth. f (x) = ab x + d.

46. −50 = −  1 _ 2  −x

47. 116 = 1 _ 4  1 _ 8 

x 48. 12 = 2(3)x + 1

49. 5 = 3  1 _ 2  x − 1

− 2 50. −30 = −4(2)x + 2 + 2

exTenSIOnS

51. Explore and discuss the graphs of f (x) = (b)x and g(x) =  1 _ b 

x . Then make a conjecture about the

relationship between the graphs of the functions b x and  1 _ b 

x for any real number b > 0.

52. Prove the conjecture made in the previous exercise.

53. Explore and discuss the graphs of f (x) = 4x, g(x) = 4x − 2, and h(x) =  1 _ 16  4x. Then make a conjecture about the relationship between the graphs of the functions b x and  1 _ bn  b

x for any real number n and real number b > 0.

54. Prove the conjecture made in the previous exercise.

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SECTION 6.3 sectioN exercises 497

6 .3 SeCTIOn exeRCISeS

veRbAl

1. What is a base b logarithm? Discuss the meaning by interpreting each part of the equivalent equations b y = x and logb(x) = y for b > 0, b ≠ 1.

2. How is the logarithmic function f (x) = logb(x) related to the exponential function g(x) = b x ? What is the result of composing these two functions?

3. How can the logarithmic equation logb x = y be solved for x using the properties of exponents?

4. Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base b, and how does the notation differ?

5. Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base b, and how does the notation differ?

AlgebRAIC

For the following exercises, rewrite each equation in exponential form.

6. log4(q) = m 7. loga(b) = c 8. log16(y) = x 9. logx(64) = y

10. logy(x) = −11 11. log15(a) = b 12. logy(137) = x 13. log13(142) = a

14. log(v) = t 15. ln(w) = n

For the following exercises, rewrite each equation in logarithmic form.

16. 4x = y 17. c d = k 18. m−7 = n 19. 19x = y

20. x − 10 __ 13 = y 21. n4 = 103 22.  7 _ 5  m

= n 23. y x = 39 _ 100

24. 10a = b 25. e k = h

For the following exercises, solve for x by converting the logarithmic equation to exponential form.

26. log3(x) = 2 27. log2(x) = −3 28. log5(x) = 2 29. log3(x) = 3

30. log2(x) = 6 31. log9(x) = 1 _ 2

32. log18(x) = 2 33. log6(x) = −3

34. log(x) = 3 35. ln(x) = 2

For the following exercises, use the definition of common and natural logarithms to simplify.

36. log(1008) 37. 10 log(32) 38. 2log(0.0001) 39. e ln(1.06)

40. ln(e −5.03) 41. e ln(10.125) + 4

nUmeRIC

For the following exercises, evaluate the base b logarithmic expression without using a calculator.

42. log3  1 _ 27  43. log6( √ — 6 ) 44. log2  1 _ 8  + 4 45. 6 log8(4)

For the following exercises, evaluate the common logarithmic expression without using a calculator.

46. log(10, 000) 47. log(0.001) 48. log(1) + 7 49. 2 log(100−3)

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498 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

For the following exercises, evaluate the natural logarithmic expression without using a calculator.

50. ln  e 1 __ 3  51. ln(1) 52. ln(e−0.225) − 3 53. 25ln  e

2 __ 5 

TeChnOlOgy

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

54. log(0.04) 55. ln(15) 56. ln  4 _ 5  57. log( √ — 2 ) 58. ln( √— 2 )

exTenSIOnS

59. Is x = 0 in the domain of the function f (x) = log(x)? If so, what is the value of the function when x = 0? Verify the result.

60. Is f (x) = 0 in the range of the function f (x) = log(x)? If so, for what value of x? Verify the result.

61. Is there a number x such that ln x = 2? If so, what is that number? Verify the result.

62. Is the following true: log3(27) _

log4  1 _ 64  = −1? Verify the

result.

63. Is the following true: ln(e1.725)

_ ln(1)

= 1.725? Verify the result.

ReAl-WORld APPlICATIOnS

64. The exposure index EI for a 35 millimeter camera is a measurement of the amount of light that hits the film.

It is determined by the equation EI = log2  f 2

_ t  , where f is the “f-stop” setting on the camera, and t is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?

65. Refer to the previous exercise. Suppose the light meter on a camera indicates an EI of −2, and the desired exposure time is 16 seconds. What should the f-stop setting be?

66. The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula

log I1 _ I2

= M1 − M2

where M is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.[23] How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

23 http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014.

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SECTION 6.4 sectioN exercises 513

6 .4 SeCTIOn exeRCISeS

veRbAl 1. The inverse of every logarithmic function is an

exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

2. What type(s) of translation(s), if any, affect the range of a logarithmic function?

3. What type(s) of translation(s), if any, affect the domain of a logarithmic function?

4. Consider the general logarithmic function f (x) = logb(x). Why can’t x be zero?

5. Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

AlgebRAIC For the following exercises, state the domain and range of the function.

6. f (x) = log3(x + 4) 7. h(x) = ln  1 _ 2 − x  8. g(x) = log5(2x + 9) − 2 9. h(x) = ln(4x + 17) − 5 10. f (x) = log2(12 − 3x) − 3

For the following exercises, state the domain and the vertical asymptote of the function. 11. f (x) = logb(x − 5) 12. g(x) = ln(3 − x) 13. f (x) = log(3x + 1)

14. f (x) = 3log(−x) + 2 15. g(x) = −ln(3x + 9) − 7

For the following exercises, state the domain, vertical asymptote, and end behavior of the function. 16. f (x) = ln(2 − x) 17. f (x) = log  x − 3 _ 7  18. h(x) = −log(3x − 4) + 3 19. g(x) = ln(2x + 6) − 5 20. f (x) = log3(15 − 5x) + 6

For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.

21. h(x) = log4(x − 1) + 1 22. f (x) = log(5x + 10) + 3 23. g(x) = ln(−x) − 2

24. f (x) = log2(x + 2) − 5 25. h(x) = 3ln(x) − 9

gRAPhICAl For the following exercises, match each function in Figure 17 with the letter corresponding to its graph.

2

1

0 1

–1

–2

2

A

B

C D E

3 x

y

Figure 17

26. d(x) = log(x)

27. f (x) = ln(x)

28. g(x) = log2(x)

29. h(x) = log5(x)

30. j(x) = log25(x)

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514 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

For the following exercises, match each function in Figure 18 with the letter corresponding to its graph.

x

y

A

B

C

–1–2–3–4–5 –1 –2 –3 –4 –5 –6

1 2 3 4 5 6

31 4 52

Figure 18

31. f (x) = log 1 _ 3

(x)

32. g(x) = log2(x)

33. h(x) = log 3 _ 4

(x)

For the following exercises, sketch the graphs of each pair of functions on the same axis.

34. f (x) = log(x) and g(x) = 10x 35. f (x) = log(x) and g(x) = log 1 _ 2

(x)

36. f (x) = log4(x) and g(x) = ln(x) 37. f (x) = e x and g(x) = ln(x)

For the following exercises, match each function in Figure 19 with the letter corresponding to its graph.

x

y

AB

C –2–3–4–5 –1

–2 –3 –4 –5

1 2 3 4 5

321 4 5–1

Figure 19

38. f (x) = log4(−x + 2) 39. g(x) = −log4(x + 2) 40. h(x) = log4(x + 2)

For the following exercises, sketch the graph of the indicated function.

41. f (x) = log2(x + 2) 42. f (x) = 2log(x) 43. f (x) = ln(−x)

44. g(x) = log(4x + 16) + 4 45. g(x) = log(6 − 3x) + 1 46. h(x) = − 1 _ 2 ln(x + 1) − 3

For the following exercises, write a logarithmic equation corresponding to the graph shown.

47. Use y = log2(x) as the parent function.

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1 2 3 4 5

321 4 5

48. Use f (x) = log3(x) as the parent function.

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1 2 3 4 5

321 4 5

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SECTION 6.4 sectioN exercises 515

49. Use f (x) = log4(x) as the parent function.

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1 2 3 4 5

321 4 5

50. Use f (x) = log5(x) as the parent function.

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1 2 3 4 5

321 4 5

TeChnOlOgy

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

51. log(x − 1) + 2 = ln(x − 1) + 2 52. log(2x − 3) + 2 = −log(2x − 3) + 5 53. ln(x − 2) = −ln(x + 1)

54. 2ln(5x + 1) = 1 _ 2 ln(−5x) + 1 55. 1 _ 3 log(1 − x) = log(x + 1) +

1 _ 3

exTenSIOnS 56. Let b be any positive real number such that b ≠ 1.

What must logb1 be equal to? Verify the result. 57. Explore and discuss the graphs of f (x) = log

1 _ 2 (x)

and g(x) = −log2(x). Make a conjecture based on the result.

58. Prove the conjecture made in the previous exercise. 59. What is the domain of the function

f (x) = ln  x + 2 _ x − 4  ? Discuss the result.

60. Use properties of exponents to find the x-intercepts of the function f (x) = log(x 2 + 4x + 4) algebraically. Show the steps for solving, and then verify the result by graphing the function.

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SECTION 6.5 sectioN exercises 525

6.5 SeCTIOn exeRCISeS

veRbAl 1. How does the power rule for logarithms help when

solving logarithms with the form logb( n √

— x )?

2. What does the change-of-base formula do? Why is it useful when using a calculator?

AlgebRAIC For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

3. logb(7x · 2y) 4. ln(3ab · 5c) 5. logb  13 _ 17  6. log4 

x __ z _ w  7. ln  1 _ 4k  8. log2(yx) For the following exercises, condense to a single logarithm if possible.

9. ln(7) + ln(x) + ln(y) 10. log3(2) + log3(a) + log3(11) + log3(b) 11. logb(28) − logb(7)

12. ln(a) − ln(d) − ln(c) 13. −logb  1 _ 7  14. 1 _ 3 ln(8)

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

15. log  x15 y13 _ z19  16. ln  a −2 _

b−4 c5  17. log( √— x3 y−4 ) 18. ln  y √

_____

y _ 1 − y  19. log(x 2 y 3

3 √ —

x2 y5 )

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. 20. log(2×4) + log(3×5) 21. ln(6×9) − ln(3×2) 22. 2log(x) + 3log(x + 1)

23. log(x) − 1 _ 2 log(y) + 3log(z) 24. 4log7 (c) +

log7(a) _ 3 + log7(b) _ 3

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base. 25. log7(15) to base e 26. log14(55.875) to base 10

For the following exercises, suppose log5 (6) = a and log5 (11) = b. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a and b. Show the steps for solving.

27. log11(5) 28. log6(55) 29. log11  6 _ 11  nUmeRIC For the following exercises, use properties of logarithms to evaluate without using a calculator.

30. log3  1 _ 9  − 3log3 (3) 31. 6log8(2) + log8(64) _ 3log8(4)

32. 2log9(3) − 4log9(3) + log9  1 _ 729  For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

33. log3(22) 34. log8(65) 35. log6(5.38) 36. log4  15 _ 2  37. log 1 _ 2 (4.7)

exTenSIOnS 38. Use the product rule for logarithms to find all x

values such that log12(2x + 6) + log12(x + 2) = 2. Show the steps for solving.

39. Use the quotient rule for logarithms to find all x values such that log6(x + 2) − log6 (x − 3) = 1. Show the steps for solving.

40. Can the power property of logarithms be derived from the power property of exponents using the equation b x = m? If not, explain why. If so, show the derivation.

41. Prove that logb (n) = 1 _

logn(b) for any positive integers

b > 1 and n > 1.

42. Does log81(2401) = log3(7)? Verify the claim algebraically.

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SECTION 6.6 sectioN exercises 535

6.6 SeCTIOn exeRCISeS

veRbAl

1. How can an exponential equation be solved? 2. When does an extraneous solution occur? How can an extraneous solution be recognized?

3. When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

AlgebRAIC For the following exercises, use like bases to solve the exponential equation.

4. 4−3v − 2 = 4−v 5. 64 ⋅ 43x = 16 6. 32x + 1 ⋅ 3x = 243

7. 2−3n ⋅ 1 _ 4 = 2 n + 2 8. 625 ⋅ 53x + 3 = 125 9.

363b _ 362b

= 216 2 − b

10.  1 _ 64  3n

⋅ 8 = 26

For the following exercises, use logarithms to solve. 11. 9x − 10 = 1 12. 2e 6x = 13 13. e r + 10 − 10 = −42

14. 2 ⋅ 109a = 29 15. −8 ⋅ 10 p + 7 − 7 = −24 16. 7e 3n − 5 + 5 = −89

17. e −3k + 6 = 44 18. −5e 9x − 8 − 8 = −62 19. −6e 9x + 8 + 2 = −74 20. 2x + 1 = 52x − 1 21. e 2x − e x − 132 = 0 22. 7e8x + 8 − 5 = −95

23. 10e 8x + 3 + 2 = 8 24. 4e 3x + 3 − 7 = 53 25. 8e−5x − 2 − 4 = −90

26. 32x + 1 = 7x − 2 27. e 2x − e x − 6 = 0 28. 3e 3 − 3x + 6 = −31

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.

29. log  1 _ 100  = −2 30. log324(18) = 1 _ 2

For the following exercises, use the definition of a logarithm to solve the equation.

31. 5log7(n) = 10 32. −8log9(x) = 16 33. 4 + log2(9k) = 2 34. 2log(8n + 4) + 6 = 10 35. 10 − 4ln(9 − 8x) = 6

For the following exercises, use the one-to-one property of logarithms to solve. 36. ln(10 − 3x) = ln(−4x) 37. log13(5n − 2) = log13(8 − 5n) 38. log(x + 3) − log(x) = log(74)

39. ln(−3x) = ln(x2 − 6x) 40. log4(6 − m) = log43(m) 41. ln(x − 2) − ln(x) = ln(54)

42. log9(2n 2 − 14n)= log9(−45 + n

2) 43. ln(x2 − 10) + ln(9) = ln(10)

For the following exercises, solve each equation for x. 44. log(x + 12) = log(x) + log(12) 45. ln(x) + ln(x − 3) = ln(7x) 46. log2(7x + 6) = 3 47. ln(7) + ln(2 − 4×2) = ln(14) 48. log8(x + 6) − log8(x) = log8(58) 49. ln(3) − ln(3 − 3x) = ln(4) 50. log3(3x) − log3(6) = log3(77)

gRAPhICAl For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

51. log9(x) − 5 = −4 52. log3(x) + 3 = 2 53. ln(3x) = 2

54. ln(x − 5) = 1 55. log(4) + log(−5x) = 2 56. −7 + log3 (4 − x) = −6 57. ln(4x − 10) − 6 = −5 58. log(4 − 2x) = log(−4x) 59. log11(−2x

2 − 7x) = log11(x − 2)

60. ln(2x + 9) = ln(−5x) 61. log9(3 − x) = log9(4x − 8) 62. log(x 2 + 13) = log(7x + 3)

63. 3 _ log2(10)

− log(x − 9) = log(44) 64. ln(x) − ln(x + 3) = ln(6)

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536 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

65. An account with an initial deposit of $6,500 earns 7.25% annual interest, compounded continuously. How much will the account be worth after 20 years?

66. The formula for measuring sound intensity in decibels D is defined by the equation D = 10 log  I __ I0  , where I is the intensity of the sound in watts per square meter and I0 = 10

−12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of 8.3 ⋅ 102 watts per square meter?

67. The population of a small town is modeled by the equation P = 1650e 0.5t where t is measured in years. In approximately how many years will the town’s population reach 20,000?

TeChnOlOgy

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.

68. 1000(1.03)t = 5000 using the common log. 69. e5x = 17 using the natural log 70. 3(1.04)3t = 8 using the common log 71. 34x − 5 = 38 using the common log 72. 50e−0.12t = 10 using the natural log

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

73. 7e3x − 5 + 7.9 = 47 74. ln(3) + ln(4.4x + 6.8) = 2 75. log(−0.7x − 9) = 1 + 5log(5)

76. Atmospheric pressure P in pounds per square inch is represented by the formula P = 14.7e−0.21x, where x is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? (Hint: there are 5280 feet in a mile)

77. The magnitude M of an earthquake is represented by the equation M = 2 _ 3 log  E __ E0  where E is the amount of energy released by the earthquake in joules and E0 = 10

4.4 is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing 1.4 · 1013 joules of energy?

exTenSIOnS

78. Use the definition of a logarithm along with the one- to-one property of logarithms to prove that blogb x = x.

79. Recall the formula for continually compounding interest, y = Ae kt. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm.

80. Recall the compound interest formula A = a  1 + r _ k  kt.

Use the definition of a logarithm along with properties of logarithms to solve the formula for time t.

81. Newton’s Law of Cooling states that the temperature T of an object at any time t can be described by the equation T = Ts + (T0 − Ts)e

−kt, where Ts is the temperature of the surrounding environment, T0 is the initial temperature of the object, and k is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm.

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SECTION 6.7 sectioN exercises 549

6.7 SeCTIOn exeRCISeS

veRbAl 1. With what kind of exponential model would half-life

be associated? What role does half-life play in these models?

2. What is carbon dating? Why does it work? Give an example in which carbon dating would be useful.

3. With what kind of exponential model would doubling time be associated? What role does doubling time play in these models?

4. Define Newton’s Law of Cooling. Then name at least three real-world situations where Newton’s Law of Cooling would be applied.

5. What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.

nUmeRIC

6. The temperature of an object in degrees Fahrenheit after t minutes is represented by the equation T(t) = 68e −0.0174t + 72. To the nearest degree, what is the temperature of the object after one and a half hours?

For the following exercises, use the logistic growth model f (x) = 150 _ 1 + 8e−2x

.

7. Find and interpret f (0). Round to the nearest tenth. 8. Find and interpret f (4). Round to the nearest tenth. 9. Find the carrying capacity. 10. Graph the model.

11. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.

12. x –2 –1 0 1 2 3 4 5

f (x) 0.694 0.833 1 1.2 1.44 1.728 2.074 2.488

13. Rewrite f (x) = 1.68(0.65)x as an exponential equation with base e to five significant digits.

TeChnOlOgy For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.

14. x 1 2 3 4 5 6 7 8 9 10

f (x) 2 4.079 5.296 6.159 6.828 7.375 7.838 8.238 8.592 8.908

15. x 1 2 3 4 5 6 7 8 9 10

f (x) 2.4 2.88 3.456 4.147 4.977 5.972 7.166 8.6 10.32 12.383 16. x 4 5 6 7 8 9 10 11 12 13

f (x) 9.429 9.972 10.415 10.79 11.115 11.401 11.657 11.889 12.101 12.295 17.

x 1.25 2.25 3.56 4.2 5.65 6.75 7.25 8.6 9.25 10.5

f (x) 5.75 8.75 12.68 14.6 18.95 22.25 23.75 27.8 29.75 33.5

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in t years is modeled by the equation P(t) = 1000 _

1 + 9e−0.6t .

18. Graph the function. 19. What is the initial population of fish? 20. To the nearest tenth, what is the doubling time for

the fish population? 21. To the nearest whole number, what will the fish

population be after 2 years? 22. To the nearest tenth, how long will it take for the

population to reach 900? 23. What is the carrying capacity for the fish population?

Justify your answer using the graph of P.

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550 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

exTenSIOnS 24. A substance has a half-life of 2.045 minutes. If the

initial amount of the substance was 132.8 grams, how many half-lives will have passed before the substance decays to 8.3 grams? What is the total time of decay?

25. The formula for an increasing population is given by P(t) = P0e

rt where P0 is the initial population and r > 0. Derive a general formula for the time t it takes for the population to increase by a factor of M.

26. Recall the formula for calculating the magnitude of an earthquake, M = 2 _ 3 log  S __ S0  . Show each step for solving this equation algebraically for the seismic moment S.

27. What is the y-intercept of the logistic growth model y = c ________ 1 + ae−rx ? Show the steps for calculation. What does this point tell us about the population?

28. Prove that b x = e xln(b) for positive b ≠ 1.

ReAl-WORld APPlICATIOnS For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.

29. To the nearest hour, what is the half-life of the drug? 30. Write an exponential model representing the amount of the drug remaining in the patient’s system after t hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 3 hours. Round to the nearest milligram.

31. Using the model found in the previous exercise, find f (10) and interpret the result. Round to the nearest hundredth.

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of 1.15% per day.

32. To the nearest day, how long will it take for half of the Iodine-125 to decay?

33. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after t days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.

34. A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?

35. The half-life of Radium-226 is 1590 years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

36. The half-life of Erbium-165 is 10.4 hours. What is the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

37. A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is 5730 years.)

38. A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?

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SECTION 6.7 sectioN exercises 551

For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1,000 bacteria present after 20 minutes.

39. To the nearest whole number, what was the initial population in the culture?

40. Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?

For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of 100° Fahrenheit was taken off the stove to cool in a 69° F room. After fifteen minutes, the internal temperature of the soup was 95° F.

41. Use Newton’s Law of Cooling to write a formula that models this situation.

42. To the nearest minute, how long will it take the soup to cool to 80° F?

43. To the nearest degree, what will the temperature be after 2 and a half hours?

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of 165° Fahrenheit and is allowed to cool in a 75° F room. After half an hour, the internal temperature of the turkey is 145° F.

44. Write a formula that models this situation. 45. To the nearest degree, what will the temperature be after 50 minutes?

46. To the nearest minute, how long will it take the turkey to cool to 110° F?

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.

47.

0–1–2–3–4–5 1 2 3 4

log (x)

5

48.

0–1–2–3–4–5 1 2 3 4

log (x)

5

49. Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: 10−10 W ___ m2 ,

Vacuum: 10−4 W ___ m2 , Jet: 10 2 W ___ m2

50. Recall the formula for calculating the magnitude of an earthquake, M = 2 __ 3 log  S __ S0  . One earthquake has magnitude 3.9 on the MMS scale. If a second earthquake has 750 times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.

For the following exercises, use this scenario: The equation N(t) = 500 _ 1 + 49e−0.7t

models the number of people in a town who have heard a rumor after t days.

51. How many people started the rumor? 52. To the nearest whole number, how many people will have heard the rumor after 3 days?

53. As t increases without bound, what value does N(t) approach? Interpret your answer.

For the following exercise, choose the correct answer choice.

54. A doctor and injects a patient with 13 milligrams of radioactive dye that decays exponentially. After 12 minutes, there are 4.75 milligrams of dye remaining in the patient’s system. Which is an appropriate model for this situation?

a. f (t) = 13(0.0805)t b. f (t) = 13e0.9195t c. f (t) = 13e(−0.0839t) d. f (t) = 4.75 __________ 1 + 13e−0.83925t

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SECTION 6.8 sectioN exercises 561

6.8 SeCTIOn exeRCISeS

veRbAl

1. What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

2. What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

3. What is regression analysis? Describe the process of performing regression analysis on a graphing utility.

4. What might a scatterplot of data points look like if it were best described by a logarithmic model?

5. What does the y-intercept on the graph of a logistic equation correspond to for a population modeled by that equation?

gRAPhICAl

For the following exercises, match the given function of best fit with the appropriate scatterplot in Figure 7 through Figure 11. Answer using the letter beneath the matching graph.

1

2 4 6 8

10 12 14 16

y

x 2 3 4 5 6 7 8 9 10

(a)

Figure 7

1

2 4 6 8

10 12 14 16

y

x 2 3 4 5 6 7 8 9 10

(b)

Figure 8

2 4 6 8

10 12 14 16

y

1 2 3 4 5 6 7 8 9 10 x

(c)

Figure 9

1

2 4 6 8

10 12 14 16

y

x 2 3 4 5 6 7 8 9 10

(d)

Figure 10

1

2 4 6 8

10 12 14 16

y

x 2 3 4 5 6 7 8 9 10

(e)

Figure 11

6. y = 10.209e−0.294x 7. y = 5.598 − 1.912ln(x) 8. y = 2.104(1.479)x

9. y = 4.607 + 2.733ln(x) 10. y = 14.005 __________ 1 + 2.79e−0.812x

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562 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

nUmeRIC

11. To the nearest whole number, what is the initial value of a population modeled by the logistic equation P(t) = 175 ____________ 1 + 6.995e−0.68t ? What is the carrying capacity?

12. Rewrite the exponential model A(t) = 1550(1.085)x as an equivalent model with base e. Express the exponent to four significant digits.

13. A logarithmic model is given by the equation h(p) = 67.682 − 5.792ln(p). To the nearest hundredth, for what value of p does h(p) = 62?

14. A logistic model is given by the equation P(t) = 90 ________ 1 + 5e−0.42t . To the nearest hundredth, for

what value of t does P(t) = 45?

15. What is the y-intercept on the graph of the logistic model given in the previous exercise?

TeChnOlOgy For the following exercises, use this scenario: The population P of a koi pond over x months is modeled by the function P(x) = 68 __

1 + 16e−0.28x .

16. Graph the population model to show the population over a span of 3 years.

17. What was the initial population of koi?

18. How many koi will the pond have after one and a half years?

19. How many months will it take before there are 20 koi in the pond?

20. Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity.

For the following exercises, use this scenario: The population P of an endangered species habitat for wolves is modeled by the function P(x) = 558 __

1 + 54.8e−0.462x , where x is given in years.

21. Graph the population model to show the population over a span of 10 years.

22. What was the initial population of wolves transported to the habitat?

23. How many wolves will the habitat have after 3 years? 24. How many years will it take before there are 100 wolves in the habitat?

25. Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity.

For the following exercises, refer to Table 7.

x 1 2 3 4 5 6 f (x) 1125 1495 2310 3294 4650 6361

Table 7

26. Use a graphing calculator to create a scatter diagram of the data.

27. Use the regression feature to find an exponential function that best fits the data in the table.

28. Write the exponential function as an exponential equation with base e.

29. Graph the exponential equation on the scatter diagram.

30. Use the intersect feature to find the value of x for which f (x) = 4000.

SECTION 6.8 sectioN exercises 563

For the following exercises, refer to Table 8.

x 1 2 3 4 5 6 f (x) 555 383 307 210 158 122

Table 8

31. Use a graphing calculator to create a scatter diagram of the data.

32. Use the regression feature to find an exponential function that best fits the data in the table.

33. Write the exponential function as an exponential equation with base e.

34. Graph the exponential equation on the scatter diagram.

35. Use the intersect feature to find the value of x for which f (x) = 250.

For the following exercises, refer to Table 9.

x 1 2 3 4 5 6 f (x) 5.1 6.3 7.3 7.7 8.1 8.6

Table 9

36. Use a graphing calculator to create a scatter diagram of the data.

37. Use the LOGarithm option of the REGression feature to find a logarithmic function of the form y = a + bln(x) that best fits the data in the table.

38. Use the logarithmic function to find the value of the function when x = 10.

39. Graph the logarithmic equation on the scatter diagram.

40. Use the intersect feature to find the value of x for which f (x) = 7.

For the following exercises, refer to Table 10.

x 1 2 3 4 5 6 7 8 f (x) 7.5 6 5.2 4.3 3.9 3.4 3.1 2.9

Table 10

41. Use a graphing calculator to create a scatter diagram of the data.

42. Use the LOGarithm option of the REGression feature to find a logarithmic function of the form y = a + bln(x) that best fits the data in the table.

43. Use the logarithmic function to find the value of the function when x = 10.

44. Graph the logarithmic equation on the scatter diagram.

45. Use the intersect feature to find the value of x for which f (x) = 8.

For the following exercises, refer to Table 11.

x 1 2 3 4 5 6 7 8 9 10 f (x) 8.7 12.3 15.4 18.5 20.7 22.5 23.3 24 24.6 24.8

Table 11

46. Use a graphing calculator to create a scatter diagram of the data.

47. Use the LOGISTIC regression option to find a logistic growth model of the form y = c _

1 + ae−b x that

best fits the data in the table.

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564 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

48. Graph the logistic equation on the scatter diagram. 49. To the nearest whole number, what is the predicted carrying capacity of the model?

50. Use the intersect feature to find the value of x for which the model reaches half its carrying capacity.

For the following exercises, refer to Table 12.

x 0 2 4 5 7 8 10 11 15 17 f (x) 12 28.6 52.8 70.3 99.9 112.5 125.8 127.9 135.1 135.9

Table 12

51. Use a graphing calculator to create a scatter diagram of the data.

52. Use the LOGISTIC regression option to find a logistic growth model of the form y = c ________ 1 + ae−b x that best fits the data in the table.

53. Graph the logistic equation on the scatter diagram. 54. To the nearest whole number, what is the predicted carrying capacity of the model?

55. Use the intersect feature to find the value of x for which the model reaches half its carrying capacity.

exTenSIOnS

56. Recall that the general form of a logistic equation for a population is given by P(t) = c _

1 + ae−bt ,

such that the initial population at time t = 0 is P(0) = P0. Show algebraically that

c − P(t)

_ P(t)

= c − P0 _

P0 e−bt.

57. Use a graphing utility to find an exponential regression formula f (x) and a logarithmic regression formula g(x) for the points (1.5, 1.5) and (8.5, 8.5). Round all numbers to 6 decimal places. Graph the points and both formulas along with the line y = x on the same axis. Make a conjecture about the relationship of the regression formulas.

58. Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary.

59. Find the inverse function f −1 (x) for the logistic function f (x) = c _

1 + ae−b x . Show all steps.

60. Use the result from the previous exercise to graph the logistic model P(t) = 20 ________ 1 + 4e−0.5t along with its

inverse on the same axis. What are the intercepts and asymptotes of each function?

CHAPTER 6 review 565

ChAPTeR 6 RevIeW

Key Terms annual percentage rate (APR) the yearly interest rate earned by an investment account, also called nominal rate

carrying capacity in a logistic model, the limiting value of the output

change-of-base formula a formula for converting a logarithm with any base to a quotient of logarithms with any other base.

common logarithm the exponent to which 10 must be raised to get x; log10(x) is written simply as log(x).

compound interest interest earned on the total balance, not just the principal

doubling time the time it takes for a quantity to double

exponential growth a model that grows by a rate proportional to the amount present

extraneous solution a solution introduced while solving an equation that does not satisfy the conditions of the original equation

half-life the length of time it takes for a substance to exponentially decay to half of its original quantity

logarithm the exponent to which b must be raised to get x; written y = logb(x)

logistic growth model a function of the form f (x) = c ________ 1+ ae−b x where c _ 1 + a is the initial value, c is the carrying capacity, or

limiting value, and b is a constant determined by the rate of growth

natural logarithm the exponent to which the number e must be raised to get x; loge(x) is written as ln(x).

Newton’s Law of Cooling the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature

nominal rate the yearly interest rate earned by an investment account, also called annual percentage rate

order of magnitude the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal

power rule for logarithms a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base

product rule for logarithms a rule of logarithms that states that the log of a product is equal to a sum of logarithms

quotient rule for logarithms a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms

Key equations definition of the exponential function f (x) = b x, where b > 0, b ≠ 1

definition of exponential growth f (x) = ab x, where a > 0, b > 0, b ≠ 1

compound interest formula A = a  1 + r _ k  kt, where

A(t) is the account value at time t t is the number of years P is the initial investment, often called the principal r is the annual percentage rate (APR), or nominal rate n is the number of compounding periods in one year

continuous growth formula A(t) = ae rt, where t is the number of unit time periods of growth a is the starting amount (in the continuous compounding formula a is

replaced with P, the principal) e is the mathematical constant, e ≈ 2.718282

General Form for the Translation of the f (x) = ab x + c + d Parent Function f (x) = b x

Definition of the logarithmic function For x > 0, b > 0, b ≠ 1, y = logb(x) if and only if b y = x.

Definition of the common logarithm For x > 0, y = log(x) if and only if 10y = x.

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566 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

Definition of the natural logarithm For x > 0, y = ln(x) if and only if ey = x.

General Form for the Translation of the f (x) = alogb(x + c) + d Parent Logarithmic Function f (x) = logb(x)

The Product Rule for Logarithms logb(MN) = logb(M) + logb(N)

The Quotient Rule for Logarithms logb  M __ N  = logbM − logbN The Power Rule for Logarithms logb(M

n) = nlogbM

The Change-of-Base Formula logbM = lognM _ lognb

n > 0, n ≠ 1, b ≠ 1

One-to-one property for exponential functions For any algebraic expressions S and T and any positive real number

b, where bS = bT if and only if S = T.

Definition of a logarithm For any algebraic expression S and positive real numbers b and c, where b ≠ 1,

logb(S) = c if and only if b c = S.

One-to-one property for logarithmic functions For any algebraic expressions S and T and any positive real number

b, where b ≠ 1,

logbS = logbT if and only if S = T.

Half-life formula If A = A0e kt, k < 0, the half-life is t = −

ln(2) _ k

. t = ln  A _ A0  _

−0.000121

Carbon-14 dating A0 is the amount of carbon-14 when the plant or animal died,

A is the amount of carbon-14 remaining today, t is the age of the fossil in years

Doubling time formula If A = A0e kt, k > 0, the doubling time is t = ln(2) ___ k

Newton’s Law of Cooling T(t) = Ae kt + Ts, where Ts is the ambient temperature, A = T(0) − Ts, and k is the continuous rate of cooling.

Key Concepts

6.1 Exponential Functions • An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent.

See Example 1. • A function is evaluated by solving at a specific value. See Example 2 and Example 3. • An exponential model can be found when the growth rate and initial value are known. See Example 4. • An exponential model can be found when the two data points from the model are known. See Example 5. • An exponential model can be found using two data points from the graph of the model. See Example 6. • An exponential model can be found using two data points from the graph and a calculator. See Example 7. • The value of an account at any time t can be calculated using the compound interest formula when the principal,

annual interest rate, and compounding periods are known. See Example 8. • The initial investment of an account can be found using the compound interest formula when the value of the

account, annual interest rate, compounding periods, and life span of the account are known. See Example 9. • The number e is a mathematical constant often used as the base of real world exponential growth and decay

models. Its decimal approximation is e ≈ 2.718282. • Scientific and graphing calculators have the key [e x] or [exp(x)] for calculating powers of e. See Example 10. • Continuous growth or decay models are exponential models that use e as the base. Continuous growth and decay

models can be found when the initial value and growth or decay rate are known. See Example 11 and Example 12.

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6.2 Graphs of Exponential Functions • The graph of the function f (x) = b x has a y-intercept at (0, 1), domain (−∞, ∞), range (0, ∞), and horizontal

asymptote y = 0. See Example 1. • If b > 1, the function is increasing. The left tail of the graph will approach the asymptote y = 0, and the right tail

will increase without bound. • If 0 < b < 1, the function is decreasing. The left tail of the graph will increase without bound, and the right tail

will approach the asymptote y = 0. • The equation f (x) = b x + d represents a vertical shift of the parent function f (x) = b x. • The equation f (x) = b x + c represents a horizontal shift of the parent function f (x) = b x. See Example 2. • Approximate solutions of the equation f (x) = b x + c + d can be found using a graphing calculator. See Example 3. • The equation f (x) = ab x, where a > 0, represents a vertical stretch if ∣ a ∣ > 1 or compression if 0 < ∣ a ∣ < 1 of the

parent function f (x) = b x. See Example 4. • When the parent function f (x) = b x is multiplied by −1, the result, f (x) = −b x, is a reflection about the x-axis.

When the input is multiplied by −1, the result, f (x) = b−x, is a reflection about the y-axis. See Example 5. • All translations of the exponential function can be summarized by the general equation f (x) = ab x + c + d. See

Table 3. • Using the general equation f (x) = ab x + c + d, we can write the equation of a function given its description. See

Example 6.

6.3 Logarithmic Functions • The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an

exponential function. • Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm.

See Example 1. • Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm

See Example 2. • Logarithmic functions with base b can be evaluated mentally using previous knowledge of powers of b. See

Example 3 and Example 4. • Common logarithms can be evaluated mentally using previous knowledge of powers of 10. See Example 5. • When common logarithms cannot be evaluated mentally, a calculator can be used. See Example 6. • Real-world exponential problems with base 10 can be rewritten as a common logarithm and then evaluated using

a calculator. See Example 7. • Natural logarithms can be evaluated using a calculator Example 8.

6.4 Graphs of Logarithmic Functions • To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and

solve for x. See Example 1 and Example 2. • The graph of the parent function f (x) = logb(x) has an x-intercept at (1, 0), domain (0, ∞), range (−∞, ∞),

vertical asymptote x = 0, and • if b > 1, the function is increasing. • if 0 < b < 1, the function is decreasing. See Example 3.

• The equation f (x) = logb(x + c) shifts the parent function y = logb(x) horizontally • left c units if c > 0. • right c units if c < 0. See Example 4.

• The equation f (x) = logb(x) + d shifts the parent function y = logb(x) vertically • up d units if d > 0. • down d units if d < 0. See Example 5.

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568 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

• For any constant a > 0, the equation f (x) = alogb(x) • stretches the parent function y = logb(x) vertically by a factor of a if ∣ a ∣ > 1. • compresses the parent function y = logb(x) vertically by a factor of a if ∣ a ∣ < 1. See Example 6 and Example 7.

• When the parent function y = logb(x) is multiplied by −1, the result is a reflection about the x-axis. When the input is multiplied by −1, the result is a reflection about the y-axis. • The equation f (x) = −logb(x) represents a reflection of the parent function about the x-axis.

• The equation f (x) = logb(−x) represents a reflection of the parent function about the y-axis. See Example 8. • A graphing calculator may be used to approximate solutions to some logarithmic equations See Example 9.

• All translations of the logarithmic function can be summarized by the general equation f (x) = alogb(x + c) + d. See Table 4.

• Given an equation with the general form f (x) = alogb(x + c) + d, we can identify the vertical asymptote x = −c for the transformation. See Example 10.

• Using the general equation f (x) = alogb(x + c) + d, we can write the equation of a logarithmic function given its graph. See Example 11.

6.5 Logarithmic Properties • We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See

Example 1. • We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See

Example 2. • We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the

log of its base. See Example 3, Example 4, and Example 5. • We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm

with a complex input. See Example 6, Example 7, and Example 8. • The rules of logarithms can also be used to condense sums, differences, and products with the same base as a

single logarithm. See Example 9, Example 10, Example 11, and Example 12. • We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-

base formula. See Example 13. • The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and e as the quotient

of natural or common logs. That way a calculator can be used to evaluate. See Example 14.

6.6 Exponential and Logarithmic Equations • We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with

the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.

• When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See Example 1.

• When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See Example 2, Example 3, and Example 4.

• When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See Example 5.

• We can solve exponential equations with base e, by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See Example 6 and Example 7.

• After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See Example 8.

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• When given an equation of the form logb(S) = c, where S is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation bc = S, and solve for the unknown. See Example 9 and Example 10.

• We can also use graphing to solve equations with the form logb(S) = c. We graph both equations y = logb(S) and y = c on the same coordinate plane and identify the solution as the x-value of the intersecting point. See Example 11.

• When given an equation of the form logbS = logbT, where S and T are algebraic expressions, we can use the one- to-one property of logarithms to solve the equation S = T for the unknown. See Example 12.

• Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See Example 13.

6.7 Exponential and Logarithmic Models • The basic exponential function is f (x) = ab x. If b > 1, we have exponential growth; if 0 < b < 1, we have

exponential decay. • We can also write this formula in terms of continuous growth as A = A0e

kx, where A0 is the starting value. If A0 is positive, then we have exponential growth when k > 0 and exponential decay when k < 0. See Example 1.

• In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See Example 2.

• We can find the age, t, of an organic artifact by measuring the amount, k, of carbon-14 remaining in the artifact

and using the formula t = ln(k)

_ −0.000121 to solve for t. See Example 3.

• Given a substance’s doubling time or half-life we can find a function that represents its exponential growth or decay. See Example 4.

• We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See Example 5.

• We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. See Example 6.

• We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See Example 7.

• Any exponential function with the form y = ab x can be rewritten as an equivalent exponential function with the form y = A0e

kx where k = lnb. See Example 8.

6.8 Fitting Exponential Models to Data • Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without

bound, or where decay begins rapidly and then slows down to get closer and closer to zero. • We use the command “ExpReg” on a graphing utility to fit function of the form y = ab x to a set of data points.

See Example 1. • Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then

slows over time. • We use the command “LnReg” on a graphing utility to fit a function of the form y = a + bln(x) to a set of data

points. See Example 2. • Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows

as the function approaches an upper limit. • We use the command “Logistic” on a graphing utility to fit a function of the form y = c _________ 1 + ae−b x to a set of data points. See Example 3.

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570 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

ChAPTeR 6 RevIeW exeRCISeS

exPOnenTIAl FUnCTIOnS

1. Determine whether the function y = 156(0.825)t represents exponential growth, exponential decay, or neither. Explain

2. The population of a herd of deer is represented by the function A(t) = 205(1.13)t, where t is given in years. To the nearest whole number, what will the herd population be after 6 years?

3. Find an exponential equation that passes through the points (2, 2.25) and (5, 60.75). 4. Determine whether Table 1 could represent a function that is linear, exponential, or neither. If it appears to be

exponential, find a function that passes through the points.

x 1 2 3 4 f (x) 3 0.9 0.27 0.081

Table 1

5. A retirement account is opened with an initial deposit of $8,500 and earns 8.12% interest compounded monthly. What will the account be worth in 20 years?

6. Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with 7.5% APR, compounded daily, in order to reach her goal in 3 years?

7. Does the equation y = 2.294e−0.654t represent continuous growth, continuous decay, or neither? Explain.

8. Suppose an investment account is opened with an initial deposit of $10,500 earning 6.25% interest, compounded continuously. How much will the account be worth after 25 years?

gRAPhS OF exPOnenTIAl FUnCTIOnS 9. Graph the function f (x) = 3.5(2)x. State the domain

and range and give the y-intercept. 10. Graph the function f (x) = 4  1 __ 8 

x and its reflection about the y-axis on the same axes, and give the y-intercept.

11. The graph of f (x) = 6.5x is reflected about the y-axis and stretched vertically by a factor of 7. What is the equation of the new function, g (x) ? State its y-intercept, domain, and range.

12. The graph below shows transformations of the graph of f (x) = 2x. What is the equation for the transformation?

2

x

y

–1–1–2

–2

–3

–3

–4–5–6

1

3

3

210 4

4 5 6 7 8 9

5 6

Figure 1

lOgARIThmIC FUnCTIOnS 13. Rewrite log17(4913) = x as an equivalent exponential

equation. 14. Rewrite ln(s) = t as an equivalent exponential

equation. 15. Rewrite a −

2 __ 5 = b as an equivalent logarithmic equation.

16. Rewrite e−3.5 = h as an equivalent logarithmic equation.

17. Solve for xlog64(x) =  1 _ 3  to exponential form. 18. Evaluate log5  1 _ 125  without using a calculator.

19. Evaluate log(0.000001) without using a calculator. 20. Evaluate log(4.005) using a calculator. Round to the nearest thousandth.

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21. Evaluate ln(e−0.8648) without using a calculator. 22. Evaluate ln  3 √ —

18  using a calculator. Round to the nearest thousandth.

gRAPhS OF lOgARIThmIC FUnCTIOnS 23. Graph the function g(x) = log(7x + 21) − 4. 24. Graph the function h(x) = 2ln(9 − 3x) + 1. 25. State the domain, vertical asymptote, and end

behavior of the function g (x) = ln(4x + 20) − 17.

lOgARIThmIC PROPeRTIeS

26. Rewrite ln(7r · 11st) in expanded form. 27. Rewrite log8(x) + log8(5) + log8(y) + log8(13) in compact form.

28. Rewrite logm  67 ___ 83  in expanded form. 29. Rewrite ln(z) – ln(x) – ln(y) in compact form. 30. Rewrite ln  1 __ x5  as a product. 31. Rewrite −logy 

1 __ 12  as a single logarithm. 32. Use properties of logarithms to expand log  r 2s11 _ t14  . 33. Use properties of logarithms to expand

ln  2b √ ______

b + 1 _____ b − 1  . 34. Condense the expression 5ln(b) + ln(c) +

ln(4 − a) _ 2 to a single logarithm.

35. Condense the expression 3log7v + 6log7w − log 7 u _ 3 to a single logarithm.

36. Rewrite log3(12.75) to base e. 37. Rewrite 5 12x − 17 = 125 as a logarithm. Then apply

the change of base formula to solve for x using the common log. Round to the nearest thousandth.

exPOnenTIAl And lOgARIThmIC eQUATIOnS 38. Solve 2163x · 216x = 363x + 2 by rewriting each side

with a common base. 39. Solve 125 __

 1 _ 625  −x − 3 = 5

3 by rewriting each side with a common base.

40. Use logarithms to find the exact solution for 7 · 17−9x − 7 = 49. If there is no solution, write no solution.

41. Use logarithms to find the exact solution for 3e6n − 2 + 1 = −60. If there is no solution, write no solution.

42. Find the exact solution for 5e3x − 4 = 6 . If there is no solution, write no solution.

43. Find the exact solution for 2e5x − 2 − 9 = −56. If there is no solution, write no solution.

44. Find the exact solution for 52x − 3 = 7x + 1. If there is no solution, write no solution.

45. Find the exact solution for e 2x − e x − 110 = 0. If there is no solution, write no solution.

46. Use the definition of a logarithm to solve. −5log7(10n) = 5.

47. Use the definition of a logarithm to find the exact solution for 9 + 6ln(a + 3) = 33.

48. Use the one-to-one property of logarithms to find an exact solution for log8(7) + log8(−4x) = log8(5). If there is no solution, write no solution.

49. Use the one-to-one property of logarithms to find an exact solution for ln(5) + ln(5×2 − 5) = ln(56). If there is no solution, write no solution.

50. The formula for measuring sound intensity in decibels D is defined by the equation D = 10log

 I _ I0  , where I is the intensity of the sound in watts per square meter and I0 = 10

−12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of 6.3 · 10−3 watts per square meter?

51. The population of a city is modeled by the equation P(t) = 256, 114e0.25t where t is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

52. Find the inverse function f −1 for the exponential function f (x) = 2 · e x + 1 − 5.

53. Find the inverse function f −1 for the logarithmic function f (x) = 0.25 · log2(x

3 + 1).

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572 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

exPOnenTIAl And lOgARIThmIC mOdelS For the following exercises, use this scenario: A doctor prescribes 300 milligrams of a therapeutic drug that decays by about 17% each hour.

54. To the nearest minute, what is the half-life of the drug? 55. Write an exponential model representing the amount of the drug remaining in the patient’s system after t hours.

Then use the formula to find the amount of the drug that would remain in the patient’s system after 24 hours. Round to the nearest hundredth of a gram.

For the following exercises, use this scenario: A soup with an internal temperature of 350° Fahrenheit was taken off the stove to cool in a 71°F room. After fifteen minutes, the internal temperature of the soup was 175°F.

56. Use Newton’s Law of Cooling to write a formula that models this situation.

57. How many minutes will it take the soup to cool to 85°F?

For the following exercises, use this scenario: The equation N(t) = 1200 __ 1 + 199e−0.625t

models the number of people in a school who have heard a rumor after t days.

58. How many people started the rumor? 59. To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity? 60. What is the carrying capacity?

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

61. x 1 2 3 4 5 6 7 8 9 10 f (x) 3.05 4.42 6.4 9.28 13.46 19.52 28.3 41.04 59.5 86.28

62. x 0.5 1 3 5 7 10 12 13 15 17 20 f (x) 18.05 17 15.33 14.55 14.04 13.5 13.22 13.1 12.88 12.69 12.45

63. Find a formula for an exponential equation that goes through the points (−2, 100) and (0, 4). Then express the formula as an equivalent equation with base e.

FITTIng exPOnenTIAl mOdelS TO dATA

64. What is the carrying capacity for a population modeled by the logistic equation P(t) = 250, 000 ___________ 1 + 499e−0.45t ? What is the initial population for the model?

65. The population of a culture of bacteria is modeled by the logistic equation P(t) = 14, 250 __

1 + 29e−0.62t , where t is in

days. To the nearest tenth, how many days will it take the culture to reach 75% of its carrying capacity?

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

66. x 1 2 3 4 5 6 7 8 9 10 f (x) 409.4 260.7 170.4 110.6 74 44.7 32.4 19.5 12.7 8.1

67. x 0.15 0.25 0.5 0.75 1 1.5 2 2.25 2.75 3 3.5 f (x) 36.21 28.88 24.39 18.28 16.5 12.99 9.91 8.57 7.23 5.99 4.81

68. x 0 2 4 5 7 8 10 11 15 17 f (x) 9 22.6 44.2 62.1 96.9 113.4 133.4 137.6 148.4 149.3

573CHAPTER 6 Practice test

ChAPTeR 6 PRACTICe TeST

1. The population of a pod of bottlenose dolphins is modeled by the function A(t) = 8(1.17)t, where t is given in years. To the nearest whole number, what will the pod population be after 3 years?

2. Find an exponential equation that passes through the points (0, 4) and (2, 9).

3. Drew wants to save $2,500 to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with 6.25% APR, compounding daily, in order to reach his goal in 4 years?

4. An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?

5. Graph the function f (x) = 5(0.5)−x and its reflection across the y-axis on the same axes, and give the y-intercept.

6. The graph below shows transformations of the graph of f (x) =  1 __ 2 

x. What is the equation for the transformation?

2

x

y

–1–1–2

–2

–3

–3

–4

–4 –5 –6 –7 –8

1

3

3

210 4 5 6 7 8

4

7. Rewrite log8.5(614.125) = a as an equivalent exponential equation.

8. Rewrite e 1 __ 2 = m as an equivalent logarithmic

equation. 9. Solve for x by converting the logarithmic equation

log 1 _ 7 (x) = 2 to exponential form. 10. Evaluate log(10,000,000) without using a calculator.

11. Evaluate ln(0.716) using a calculator. Round to the nearest thousandth.

12. Graph the function g (x) = log(12 − 6x) + 3.

13. State the domain, vertical asymptote, and end behavior of the function f (x) = log5(39 − 13x) + 7.

14. Rewrite log(17a · 2b) as a sum.

15. Rewrite logt(96) − logt(8) in compact form. 16. Rewrite log8  a 1 __ b  as a product.

17. Use properties of logarithm to expand ln (y 3z 2 · 3 √— x − 4 ).

18. Condense the expression

4ln(c) + ln(d) + ln(a)

_ 3 + ln(b + 3)

_ 3 to a single logarithm.

19. Rewrite 163x − 5 = 1000 as a logarithm. Then apply the change of base formula to solve for x using the natural log. Round to the nearest thousandth.

20. Solve  1 _ 81  x · 1 _ 243 = 

1 _ 9  −3x − 1

by rewriting each side with a common base.

21. Use logarithms to find the exact solution for −9e10a − 8 −5 = −41. If there is no solution, write no solution.

22. Find the exact solution for 10e 4x + 2 + 5 = 56. If there is no solution, write no solution.

23. Find the exact solution for −5e−4x − 1 − 4 = 64. If there is no solution, write no solution.

24. Find the exact solution for 2x − 3 = 62x − 1. If there is no solution, write no solution.

25. Find the exact solution for e2x − e x − 72 = 0. If there is no solution, write no solution.

26. Use the definition of a logarithm to find the exact solution for 4log(2n) − 7 = −11.

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574 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs

27. Use the one-to-one property of logarithms to find an exact solution for log(4×2 − 10) + log(3) = log(51) If there is no solution, write no solution.

28. The formula for measuring sound intensity in decibels D is defined by the equation

D = 10log  I __ I0  where I is the intensity of the sound in watts per square meter and I0 = 10

−12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of 4.7 · 10−1 watts per square meter?

29. A radiation safety officer is working with 112 grams of a radioactive substance. After 17 days, the sample has decayed to 80 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance?

30. Write the formula found in the previous exercise as an equivalent equation with base e. Express the exponent to five significant digits.

31. A bottle of soda with a temperature of 71° Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of 35° F. After ten minutes, the internal temperature of the soda was 63° F. Use Newton’s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?

32. The population of a wildlife habitat is modeled by the equation P(t) = 360 __ 1 + 6.2e−0.35t

, where t is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?

33. Enter the data from Table 2 into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

x 1 2 3 4 5 6 7 8 9 10 f (x) 3 8.55 11.79 14.09 15.88 17.33 18.57 19.64 20.58 21.42

Table 2

34. The population of a lake of fish is modeled by the logistic equation P(t) = 16, 120 __

1 + 25e−0.75t , where t is time in

years. To the nearest hundredth, how many years will it take the lake to reach 80% of its carrying capacity?

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

35. x 1 2 3 4 5 6 7 8 9 10 f (x) 20 21.6 29.2 36.4 46.6 55.7 72.6 87.1 107.2 138.1

36. x 3 4 5 6 7 8 9 10 11 12 13 f (x) 13.98 17.84 20.01 22.7 24.1 26.15 27.37 28.38 29.97 31.07 31.43

37. x 0 0.5 1 1.5 2 3 4 5 6 7 8 f (x) 2.2 2.9 3.9 4.8 6.4 9.3 12.3 15 16.2 17.3 17.9

ODD ANSWERSC-16

41. y = 3 _ 4 x 2

30

x

y

−2−4−6−8−10

15

6

45

42 8

60

10

75

−30 −15

−45 −60 −75

43. y = 1 __ 3 √ — x

4

x

y

−5−10−15−20−25

2

15

6

105 20

8

25

10

−4 −2

−6 −8

−10

45. y = 4 __ x2

4

x

y

−2−4−6−8−10

2

6

6

42 8

8

10

10

−4 −2

−6 −8

−10

47. ≈ 1.89 years 49. ≈ 0.61 years 51. 3 seconds 53. 48 inches 55. ≈ 49.75 pounds 57. ≈ 33.33 amperes 59. ≈ 2.88 inches

Chapter 5 Review exercises 1. f (x) = (x − 2)2 −9; vertex: (2, −9); intercepts: (5, 0), (−1, 0), (0, −5)

x

f(x)

−1 −2 −4 −6

−10 −8

−2−3−4−5 321 4 5 6 7

6 4 2

8 10

3. f (x) = 3 _ 25 (x + 2) 2 + 3

5. 300 meters by 150 meters, the longer side parallel to the river 7. Yes; degree: 5, leading coefficient: 4 9. Yes; degree: 4; leading coefficient: 1

11. As x → −∞, f (x) → −∞, as x → ∞, f (x) → ∞

13. −3 with multiplicity 2, − 1 _ 2 with multiplicity 1, −1 with multiplicity 3 15. 4 with multiplicity 1 17. 1 _ 2 with multiplicity 1, 3 with multiplicity 3 19. x 2 + 4 with remainder is 12 21. x 2 − 5x + 20 − 61 _ x + 3

23. 2x 2 − 2x − 3, so factored form is (x + 4)(2x 2 − 2x − 3)

25.  −2, 4, − 1 _ 2  27.  1, 3, 4, 1 _ 2 

29. 2 or 0 positive, 1 negative

31. Intercepts: (−2, 0),  0, − 2 _ 5  , asymptotes: x = 5 and y = 1

x −5

−10 −15

−25 −20

5 10 15 20 25 30−5−10−15−20−25

15 10

5

20 25

y

y = 1

x = 5

33. Intercepts: (3, 0), (−3, 0),  0, 27 _ 2  ; asymptotes: x = 1, −2 and y = 3

y

x −8

−16 −24

−40 −32

3 6 9 12 15−3−6−9−12−15

24 16

8

32 40

x = −2

y = 3

x = 1

35. y = x − 2 37. f −1(x) = √ — x + 2 39. f −1(x) = √

— x + 11 − 3

41. f −1(x) = (x + 3)2 − 5

__ 4 , x ≥ −3 43. y = 64 45. y = 72

47. ≈ 148.5 pounds

Chapter 5 practice test

1. Degree: 5, leading coefficient: −2 3. As x → −∞, f (x) → ∞, as x → ∞, f (x) → ∞ 5. f (x) = 3(x − 2)2

7. 3 with multiplicity 3, 1 _ 3 with multiplicity 1, 1with multiplicity 2

9. − 1 _ 2 with multiplicity 3, 2 with multiplicity 2

11. x 3 + 2x 2 + 7x + 14 + 26 _ x − 2 13.  −3, −1, 3 _ 2 

15. 1, −2, and − 3 _ 2 (multiplicity 2)

17. f (x) = − 2 _ 3 (x − 3) 2(x − 1)(x + 2) 19. 2 or 0 positive, 1 negative

21. (−3, 0), (1, 0),  0, 3 _ 4  ; asymptotes x = −2, 2 and y = 1

y

x −2 −4 −6

−10 −8

2 4 6 8 10−2−4−6−8−10

6 4 2

8 10

x = −2 x = 2

y = 1

23. f −1(x) = (x − 4)2 + 2, x ≥ 4

25. f −1(x) = x + 3 _ 3x − 2

27. y = 20

ChapteR 6

Section 6.1 1. Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original. 3. When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account. Thus, the annual percentage rate does not necessarily correspond to the real interest earned, which is the very definition of nominal. 5. Exponential; the population decreases by a proportional rate. 7. Not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function. 9. Forest B 11. After 20 years forest A will have 43 more trees than forest B. 13. Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but because forest B actually grows at a faster rate, the population will eventually become larger than forest A and will remain that way as long as the population growth models hold. Some factors that might influence the long-term validity of the exponential growth model are drought, an epidemic that culls the population, and other environmental and biological factors. 15. Exponential growth; the growth factor, 1.06, is greater than 1. 17. Exponential decay; the decay factor, 0.97, is between 0 and 1.

19. f (x) = 2000(0.1)x 21. f (x) =  1 __ 6  − 35–  1 __ 6 

x 5– ≈ 2.93(0.699)x

23. Linear 25. Neither 27. Linear 29. $10,250

31. $13,268.58 33. P = A(t) ⋅  1 + r _ n  −nt

35. $4,569.10 37. 4% 39. Continuous growth; the growth rate is greater than 0. 41. Continuous decay; the growth rate is less than 0.

43. $669.42 45. f (−1) = −4 47. f (−1) ≈ −0.2707 49. f (3) ≈ 483.8146 51. y = 3 ⋅ 5x 53. y ≈ 18 ⋅ 1.025x

55. y ≈ 0.2 ⋅ 1.95x

ODD ANSWERS C-17

57. APY =

A(t) − a _ a =

a  1 + r ___ 365  365(1)

− a ___ a

= a   1 + r ___ 365 

365 − 1 

___ a =  1 + r _

365 

365 − 1;

I(n) =  1 + r _ n  n − 1

59. Let f be the exponential decay function f (x) = a ⋅  1 _ b  x

such that b > 1. Then for some number n > 0, f (x) = a ⋅  1 _ b 

x = a(b−1)x = a ((en)−1 )x = a(e−n)x = a(e)−nx.

61. 47,622 foxes 63. 1.39%; $155,368.09 65. $35,838.76 67. $82,247.78; $449.75

Section 6.2 1. An asymptote is a line that the graph of a function approaches, as x either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small. 3. g(x) = 4(3)−x; y-intercept: (0, 4); domain: all real numbers; range: all real numbers greater than 0. 5. g(x) = −10x + 7; y-intercept: (0, 6); domain: all real numbers; range: all real numbers less than 7.

7. g(x) = 2  1 _ 4  x ; y-intercept: (0, 2); domain: all real numbers;

range: all real numbers greater than 0.

9. y-intercept: (0, −2)

2

x −1 −1−2

−2

−3

−3

−4

−4 −5

−5

1

3

3

21 4

4

5

5

g(−x) = −2(0.25)−x

g(x) = −2(0.25)x

y 11.

y

2

x −1

−1−2

−2

−3

−3

−4

−4 −5

−5

1

3

3

21 4

4

5

5

f (x) = 3 14( )x

g(x) = 3(2)x h(x) = 3(4)x

13. B 15. A 17. E 19. D 21. C

23. y 5

2

x −1

−1−2

−2

−3

−3

−4

−4 −5

−5

1

3

3

21 4

4

5

5

f(x) = 12(4)x

−f(x) = −12 (4)x

25. y 5

2

x −1

−1−2

−2

−3

−3

−4

−4 −5

−5

1

3

3

21 4

4

5

5

−f (x) = 4(2)x − 2

f (x) = −4(2)x + 2

27. Horizontal asymptote: h(x) = 3; domain: all real numbers; range: all real numbers strictly greater than 3.

2

x

h(x)

−1−2−3−4−5

1

3

3

21 4

4

5

5 6 7 8

29. As x → ∞, f (x) → −∞; as x → −∞, f (x) → −1  31. As x → ∞, f (x) → 2; as x → −∞, f (x) → ∞ 33. f (x) = 4x − 3 35. f (x) = 4x − 5 37. f (x) = 4−x

39. y = −2x + 3 41. y = −2(3)x + 7 43. g(6) ≈ 800.3

45. h(−7) = −58 47. x ≈ −2.953 49. x ≈ −0.222

51. The graph of g(x) =  1 _ b  x is the reflection about the y-axis

of the graph of f (x) = bx; for any real number b > 0 and function

f (x) = b x, the graph of  1 _ b  x is the reflection about the y-axis, f (−x).

53. The graphs of g(x) and h(x) are the same and are a horizontal shift to the right of the graph of f (x). For any real number n, real

number b > 0, and function f (x) = b x, the graph of  1 _ bn  bx is the horizontal shift f (x − n).

Section 6.3 1. A logarithm is an exponent. Specifically, it is the exponent to which a base b is raised to produce a given value. In the expressions given, the base b has the same value. The exponent, y, in the expression by can also be written as the logarithm, log

b x,

and the value of x is the result of raising b to the power of y. 3. Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation b y = x, and then properties of exponents can be applied to solve for x. 5. The natural logarithm is a special case of the logarithm with base b in that the natural log always has base e. Rather than notating the natural logarithm as loge(x), the notation used is ln(x). 7. ac = b 9. x y = 64 11. 15b = a 13. 13a = 142 15. e n = w 17. logc(k) = d 19. log19 (y) = x 21. logn(103) = 4 23. logy  39 _ 100  = x 25. ln(h) = k 27. x = 1 _ 8 29. x = 27 31. x = 3 33. x =

1 _ 216

35. x = e2 37. 32 39. 1.06 41. 14.125 43. 1 _ 2 45. 4 47. −3 49. −12 51. 0 53. 10 55. ≈ 2.708 57. ≈ 0.151 59. No, the function has no defined value for x = 0. To verify, suppose x = 0 is in the domain of the function f (x) = log(x). Then there is some number n such that n = log(0). Rewriting as an exponential equation gives: 10n = 0, which is impossible since no such real number n exists. Therefore, x = 0 is not the domain of the function f (x) = log(x). 61. Yes. Suppose there exists a real number, x such that ln(x) = 2. Rewriting as an exponential equation gives x = e2,

which is a real number. To verify, let x = e 2. Then, by definition,

ln(x) = ln(e2) = 2. 63. No; ln(1) = 0, so ln(e1.725)

_ ln(1)

is undefined. 65. 2

Section 6.4 1. Since the functions are inverses, their graphs are mirror images about the line y = x. So for every point (a, b) on the graph of a logarithmic function, there is a corresponding point (b, a) on the graph of its inverse exponential function. 3. Shifting the function right or left and reflecting the function about the y-axis will affect its domain. 5. No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

7. Domain:  −∞,  1 _ 2  ; range: (−∞, ∞) 9. Domain:  − 17 _ 4 , ∞  ; range: (−∞, ∞ ) 11. Domain: (5, ∞); vertical asymptote: x = 5

13. Domain:  − 1 _ 3 , ∞  ; vertical asymptote: x = − 1 _ 3

This OpenStax book is available for free at http://cnx.org/content/col11758/latest

ODD ANSWERSC-18

15. Domain: (−3, ∞); vertical asymptote: x = −3

17. Domain:  3 _ 7 , ∞  ; vertical asymptote: x = 3 _ 7 ; end behavior:

as x →  3 _ 7  + , f (x) → −∞ and as x → ∞, f (x) → ∞

19. Domain: (−3, ∞); vertical asymptote: x = −3; end behavior: as x → −3+, f (x) → −∞ and as x → ∞, f (x) → ∞ 21. Domain: (1, ∞); range: (−∞, ∞); vertical asymptote: x = 1;

x-intercept:  5 _ 4 , 0  ; y-intercept: DNE 23. Domain: (−∞, 0); range: (−∞, ∞); vertical asymptote: x = 0; x-intercept: (−e2, 0); y-intercept: DNE 25. Domain: (0, ∞); range: (−∞, ∞) vertical asymptote: x = 0; x-intercept: (e3, 0); y-intercept: DNE

27. B 29. C 31. B 33. C

35.

2

−1 −1 −2 −3 −4 −5

1

3

3

21 4

4

5 876 9 10

5

f(x) = log(x)

g(x) = log (x)l 2

x

y 37.

2

−1 −1−2

−2

−3

−3

−4

−4 −5

−5

1

3

3

21 4

4

5

5

f (x) = e x

g(x) = ln(x)

x

y 39. C

41.

2

x

f(x)

−1 −1−2

−2

−3

−3

−4

−4 −5

−5

1

3

3

21 4

4

5

5

43.

2

x

y

−1 −1−2

−2

−3−4−5−6−7−8

−3 −4 −5

1

3

21

4 5

45.

2

x

g(x)

−1 −1−2

−2

−3−4−5−6

−3 −4 −5

1

3

3

21 4

4 5

47. f (x) = log2(−(x − 1)) 49. f (x) = 3log4(x + 2) 51. x = 2 53. x ≈ 2.303 55. x ≈ −0.472

57. The graphs of f (x) = log 1 _ 2

(x) and g(x) = −log 2(x) appear to be

the same; conjecture: for any positive base b ≠ 1, logb(x) = −log 1 _ b (x).

59. Recall that the argument of a logarithmic function must be

positive, so we determine where x + 2 _ x − 4 > 0. From the graph of

the function f (x) = x + 2 _ x − 4 > 0, note that the graph lies above the

x-axis on the interval (−∞, −2) and again to the right of the vertical asymptote, that is (4, ∞). Therefore, the domain is (−∞, −2)∪(4, ∞).

Section 6.5 1. Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making

the logarithm easier to calculate. Thus, logb(x 1 _ n ) = 1 _ n logb(x).

3. logb (2) + logb (7) + logb(x) + logb(y) 5. logb(13) − logb(17) 7. −kln(4) 9. ln(7xy) 11. logb(4) 13. logb(7) 15. 15log(x) + 13log(y) − 19log(z)

17. 3 _ 2 log(x) − 2log(y) 19. 8 _ 3 log(x) +

14 _ 3 log(y) 21. ln(2x 7)

23. log  xz3 _ √— y  25. log7(15) = ln(15)

_ ln(7)

27. log11(5) = 1 _ b

29. log11  6 _ 11  = a − b _

b or a _

b − 1 31. 3

33. ≈ 2.81359 35. ≈ 0.93913 37. ≈ −2.23266 39. x = 4, By the quotient rule: log6(x + 2) − log6(x − 3) = log6  x + 2 _____ x − 3  = 1 Rewriting as an exponential equation and solving for x: 61 = x + 2 _____ x − 3

0 = x + 2 _____ x − 3 − 6

0 = x + 2 _____ x − 3 − 6(x − 3) _______ (x − 3)

0 = x + 2 − 6x + 18 _____________ x − 3

0 = x − 4 _____ x − 3

x = 4 Checking, we find that log6(4 + 2) − log6 (4 − 3) =

log6(6) − log6(1) is defined, so x = 4. 41. Let b and n be positive integers greater than 1. Then, by the

change-of-base formula, logb(n) = logn(n) _ logn(b)

= 1 _ logn(b)

.

Section 6.6 1. Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve. 3. The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base. 5. x = − 1 _ 3 7. n = −1 9. b =

6 _ 5

11. x = 10 13. No solution 15. p = log  17 _ 8  − 7

17. k = − ln(38)

_ 3 19. x = ln  38 _ 3  − 8 __ 9 21. x = ln(12)

23. x = ln  3 _ 5  − 3 __ 8 25. No solution 27. x = ln(3)

29. 10−2 = 1 _ 100 31. n = 49 33. k = 1 _

36 35. x = 9 − e _ 8

37. n = 1 39. No solution 41. No solution

43. x = ± 10 _ 3 45. x = 10 47. x = 0 49. x = 3 _ 4

51. x = 9

x

y

−1 −2 −3 −4 −5 −6 −7

1

321 4 5 6 7 8 9 10 11 12

53. x = e 2 _ 3 ≈ 2.5

2

x

y

−1 −1−2

−2 −3 −4 −5

1

3

3

21 4

4

5

5

ODD ANSWERS C-19

55. x = −5

x

y

3 4

2 1

−1−1 –1−2−3−4−5−6−7−8

−2 −3

5

57. x = e + 10 _ 4 ≈ 3.2

x

y

−1 −1 −2 −3 −4 −5 −6

1

321 4 5 6 7 8 9 10

59. No solution

2

x

y

−1 −1−2

−2

−3

−3

−4

−4 −5 −6 −7 −8

1

3

3

21 4 5 6 7 8 9 10

61. x = 11 _ 5 ≈ 2.2

2

x

y

−1 −1−2

−2

−3

−3

−4

−4 −5

−5

1

3

3

21 4

4

5

5

63. x = 101 _ 11 ≈ 9.2

2

x

y

−1 −2

−2

1

6

3

42 8

4

10 12 14

5 6

65. About $27,710.24

35,000 (20, 27710.24)

f(x) = 6500e0.0725x

30,000 25,000 20,000 15,000 10,000

5,000

−1 −1

20 4 6 8 10 12 14 16 18 20 22 24 x

y

67. About 5 years

25,000 20,000 (5, 20,000)

15,000 10,000

5,000

0 1 2 3 4 5 6 x

y 69. ≈ 0.567 71. ≈ 2.078 73. ≈ 2.2401 75. ≈ −44655.7143 77. About 5.83

79. t = ln (  y ______ A  1 _ k ) 81. t = ln (  T − Ts ______ T0 − Ts  −

1 _ k

) Section 6.7 1. Half-life is a measure of decay and is thus associated with exponential decay models. The half-life of a substance or quantity is the amount of time it takes for half of the initial amount of that substance or quantity to decay. 3. Doubling time is a measure of growth and is thus associated with exponential growth models.

The doubling time of a substance or quantity is the amount of time it takes for the initial amount of that substance or quantity to double in size. 5. An order of magnitude is the nearest power of ten by which a quantity exponentially grows. It is also an approximate position on a logarithmic scale; Sample response: Orders of magnitude are useful when making comparisons between numbers that differ by a great amount. For example, the mass of Saturn is 95 times greater than the mass of Earth. This is the same as saying that the mass of Saturn is about 102 times, or 2 orders of magnitude greater, than the mass of Earth. 7. f (0) ≈ 16.7; the amount initially present is about 16.7 units. 9. 150 11. Exponential; f (x) = 1.2x 13. Logarithmic

x

f(x)

1 2 3 4 5 6 7 8 9

10

1 2 3 4 5 6 7 8 9 100

15. Logarithmic

x

y

9

10

11

12

13

4 5 6 7 8 9 10 11 12 130

17.

200

t

P(t)

−2 −100−4−6−8−10−12−14−16−18−20

100

6

300

42 8

400

10 12 14 16 18 20

500 600 700 800 900

1,000

19. About 1.4 years 21. About 7.3 years 23. Four half-lives; 8.18 minutes

25. M = 2 _ 3 log  S _ S0  3 _ 2 M = log  S _ S0  10

3M

__ 2 =  S _ S0 

S010 3M __ 2 = S

27. Let y = b x for some non-negative real number b such that b ≠ 1. Then,

ln (y) = ln (b x) ln (y) = x ln (b) e ln(y) = e xln(b)

y = e xln(b)

29. A = 125e (−0.3567t); A ≈ 43mg 31. About 60 days 33. f (t) = 250e −0.00914t; half-life: about 76 minutes 35. r ≈ − 0.0667; hourly decay rate: about 6.67% 37. f (t) = 1350 e 0.03466t; after 3 hours; P (180) ≈ 691,200 39. f (t) = 256 e (0.068110t); doubling time: about 10 minutes 41. About 88minutes 43. T(t) = 90 e (−0.008377t) + 75, where t is in minutes 45. About 113  minutes 47. log10x = 1.5; x ≈ 31.623 49. MMS Magnitude: ≈ 5.82 51. N(3) ≈ 71 53. C

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ODD ANSWERSC-20

Section 6.8 1. Logistic models are best used for situations that have limited values. For example, populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic model best describes populations. 3. Regression analysis is the process of finding an equation that best fits a given set of data points. To perform a regression analysis on a graphing utility, first list the given points using the STAT then EDIT menu. Next graph the scatter plot using the STAT PLOT feature. The shape of the data points on the scatter graph can help determine which regression feature to use. Once this is determined, select the appropriate regression analysis command from the STAT then CALC menu. 5. The y-intercept on the graph of a logistic equation corresponds to the initial population for the population model. 7. C 9. B 11. P (0) = 22; 175 13. p ≈ 2.67 15. y-intercept: (0, 15) 17. 4 koi 19. About 6.8 months. 21. y

x

100 150

50

200 250 300 350 400 450 500 550 600

5 10 15 200

23. About 38 wolves 25. About 8.7 years 27. f (x) = 776.682 (1.426)x

29. y

x

2,000 3,000

1,000

4,000 5,000 6,000 7,000

1 2 3 4 5 6 70

31. y

x

150 200

100

250 300 350 400 450 500 550 600

1 2 3 4 5 6 70

33. f (x) = 731.92e −0.3038x

35. When f (x) = 250, x ≈ 3.6 37. y = 5.063 + 1.934 log(x) 39. y

x

5 6

4

7 8 9

10

1 2 3 4 5 6 70

41. y

x 1 2 3 4 5 6 7 8 9

10

1 2 3 4 5 6 7 80

43. f (10) ≈ 2.3 45. When f (x) = 8, x ≈ 0.82

47. f (x) = 25.081 __ 1 + 3.182e−0.545x

49. About 25

51. y

x

20 30

10

40 50 60 70 80 90

100 110 120 130 140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180

53. y

x

20 30

10

40 50 60 70 80 90

100 110 120 130 140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180

55. When f (x) = 68, x ≈ 4.9 57. f (x) = 1.034341(1.281204) x; g (x) = 4.035510; the regression curves are symmetrical about y = x, so it appears that they are inverse functions.

59. f −1(x) = ln (a) − ln  c _ x − 1  __

b

Chapter 6 Review exercises 1. Exponential decay; the growth factor, 0.825, is between 0 and 1. 3. y = 0.25(3) x 5. $42,888.18 7. Continuous decay; the growth rate is negative 9. Domain: all real numbers; range: all real numbers strictly greater than zero; y-intercept: (0, 3.5)

11. g(x) = 7(6.5)−x; y-intercept: (0, 7); domain: all real numbers; range: all real numbers greater than 0. 13. 17x = 4,913 15. loga b = −

2 _ 5 17. x = 4 19. log(0.000001) = −6 21. ln(e−0.8648) = −0.8648 23.

x

g(x) 1

−1−1−2−3−4−5 1 2 3 4 5

−2 −3 −4 −5 −6 −7 −8

25. Domain: x > −5; vertical asymptote: x = −5; end behavior: as x → −5+, f (x) → −∞ and as x → ∞, f (x) → ∞ 27. log 8(65xy)

29. ln  z _ xy  31. logy(12)

x

y

1 2 3 4 5 6 7 8 9

10

−1−1−2−3−4−5−6−7−8 1 2 3 4

ODD ANSWERS C-21

33. ln(2) + ln(b) +

ln(b + 1) − ln(b − 1) __ 2 35. log 7  v

3w 6 _ 3 √

— u 

37. x =  5 _ 3 39. x = −3 41. No solution 43. No solution

45. x = ln(11) 47. a = e4 − 3 49. x = ± 9 __ 5

51. About 5.45 years 53. f −1(x) = 3 √ —

24x − 1

55. f (t) = 300(0.83)t ; f (24) ≈ 3.43 g 57. About 45 minutes

59. About 8.5 days 61. Exponential 63. y = 4(0.2)x; y = 4e–1.609438x

65. About 7.2 days 67. Logarithmic y = 16.68718 − 9.71860ln(x)

y

x

4 6

2

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

0.5 1 1.5 2 2.5 3 3.5 40

Chapter 6 practice test 1. About 13 dolphins 3. $1,947 5. y-intercept: (0, 5)

x

y

1 2 3 4 5 6 7 8

−1−1−2−3−4−5 1 2 3 4 5

f(x) = 5(0.5)−x f(−x) = 5(0.5)−x

7. 8.5a = 614.125 9. x = 1 _ 49 11. ln(0.716) ≈ − 0.334 13. Domain: x < 3; vertical asymptote: x = 3; end behavior: as x → 3−, f (x) → −∞ and as x → −∞, f (x) → ∞

15. log t(12)

17. 3ln(y) + 2ln(z) + ln(x − 4)

_ 3

19. x = ln(1000) _______ ln(16) + 5 __

3 ≈ 2.497 21. a = ln (4) + 8 ________ 10

23. No solution 25. x = ln(9) 27. x = ± 3 √ — 3 ____ 2

29. f (t) = 112e−0.019792t; half-life: about 35 days 31. T(t) = 36 e−0.025131t + 35; T(60) ≈ 43° F

33. Logarithmic y

x

2 3

1

4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22

1 2 3 4 5 6 7 8 9 10 110

35. Exponential; y = 15.10062(1.24621)x

y

x

20 30

10

40 50 60 70 80 90

100 110 120 130 140 150

1 2 3 4 5 6 7 8 9 100

37. Logistic;

y = 18.41659 __ 1 + 7.54644 e−0.68375x

y

x

2 3

1

4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 90

ChapteR 7 Section 7.1 1.

Terminal side

Vertex Initial side

3. Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction. 5. Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

x

y

10 20 30 40 50 60 70 80 90

100

1 2 3 4 5 6 7 8 9 10 110

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  • Chapter 6. Exponential and Logarithmic Functions
    • Chapter 6. Exponential and Logarithmic Functions
      • 6.1. Exponential Functions
      • 6.2. Graphs of Exponential Functions
      • 6.3. Logarithmic Functions
      • 6.4. Graphs of Logarithmic Functions
      • 6.5. Logarithmic Properties
      • 6.6. Exponential and Logarithmic Equations
      • 6.7. Exponential and Logarithmic Models
      • Glossary
      • Key Equations
      • Key Concepts
      • Review Exercises
      • Practice Test
  • Odd Answers
    • Odd Answers
      • Chapter 6

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