Chapter 3

2. Chapter 3 Exponential, Logistic, and Logarithmic Functions

3. 3.1 Exponential and Logistic Functions

4. Quick Review

5. Quick Review Solutions

6. What you’ll learn about • Exponential Functions and Their Graphs • The Natural Base e • Logistic Functions and Their Graphs • Population Models … and why Exponential and logistic functions model many growth patterns, including the growth of human and animal populations.

7. Exponential Functions

8. Exponential Functions Rules For Exponents If a > 0 and b > 0, the following hold true for all real numbers x and y.

9. Exponential Functions Use the rules for exponents to solve for x • 4x= 128 • (2)2x= 27 • 2x = 7 • x = 7/2 • 2x= 1/32 • 2x= 2-5 • x = -5

10. Exponential Functions • 27x= 9-x+1 • (33)x = (32)-x+1 • 33x= 3-2x+2 • 3x = -2x+ 2 • 5x = 2 • x = 2/5 • (x3y2/3)1/2 • x3/2y1/3

11. Example Finding an Exponential Function from its Table of Values

12. Example Finding an Exponential Function from its Table of Values

13. y 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 -2 -3 -4 -5 Exponential Functions • If b > 1, then the graph of b xwill: • Rise from left to right. • Not intersect the x-axis. • Approach the x-axis. • Have a y-intercept of (0, 1) y = 2x

14. y 5 4 3 2 1 x -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 -2 -3 -4 -5 Exponential Functions • If 0 < b < 1, then the graph of b xwill: • Fall from left to right. • Not intersect the x-axis. • Approach the x-axis. • Have a y-intercept of (0, 1) y = (1/2)x

15. Example Transforming Exponential Functions Describe how to transform the graph of f(x) = 2x into the graph g(x) = 2x-2 The graph of g(x) = 2x-2is obtained by translating the graph of f(x) = 2x by 2 units to the right.

16. Example Transforming Exponential Functions

17. Example Transforming Exponential Functions

18. The Natural Base e

19. Exponential Functions and the Base e

20. Exponential Functions and the Base e

21. Example Transforming Exponential Functions

22. Example Transforming Exponential Functions

23. Logistic Growth Functions

24. Exponential Growth and Decay

25. Exponential Functions Definitions Exponential Growth and Decay The function y = k ax, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1. y new amount yO original amount b base t time h half life

26. Exponential Functions • An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g. • Find the amount remaining after t hours. • Find the amount remaining after 60 hours. • b. y = yobt/h • y = 2 (1/2)(60/15) • y = 2(1/2)4 • y = .125 g • a. y = yobt/h • y = 2 (1/2)(t/15)

27. Exponential Functions • A bacteria double every three days. There are 50 bacteria initially present • Find the amount after 2 weeks. • When will there be 3000 bacteria? • a. y = yobt/h • y = 50 (2)(14/3) • y = 1269 bacteria

28. Exponential Functions A bacteria double every three days. There are 50 bacteria initially present When will there be 3000 bacteria? • b. y = yobt/h • 3000 = 50 (2)(t/3) • 60 = 2t/3

29. 3.2 Exponential and Logistic Modeling

30. Quick Review

31. Quick Review Solutions

32. What you’ll learn about • Constant Percentage Rate and Exponential Functions • Exponential Growth and Decay Models • Using Regression to Model Population • Other Logistic Models … and why Exponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.

33. Constant Percentage Rate Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown.

34. Exponential Population Model

35. Example Finding Growth and Decay Rates

36. Example Finding an Exponential Function Determine the exponential function with initial value = 10, increasing at a rate of 5% per year.

37. Example Modeling Bacteria Growth

38. Example Modeling Bacteria Growth

39. Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

40. Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

41. Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population.

42. Example Modeling a Rumor • A high school has 1500 students. 5 students start a rumor which spreads • logistically so that s(t) = 1500/(1 + 29 e.-.09t) models the number of • students who have heard the rumor at the end of t days, where t = 0 • is the day the rumor begins to spread • How many students have heard the rumor by the end of Day 0? • How long does it take for 1000 students to hear the rumor?

43. Example Modeling a Rumor • A high school has 1500 students. 5 students start a rumor which spreads • logistically so that s(t) = 1500/(1 + 29 e.-.09t) models the number of • students who have heard the rumor at the end of t days, where t = 0 • is the day the rumor begins to spread • How many students have heard the rumor by the end of Day 0? • How long does it take for 1000 students to hear the rumor?

44. 3.3 Logarithmic Functions and Their Graphs

45. Quick Review

46. What you’ll learn about • Inverses of Exponential Functions • Common Logarithms – Base 10 • Natural Logarithms – Base e • Graphs of Logarithmic Functions • Measuring Sound Using Decibels … and why Logarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.

47. Logarithmic Functions The inverse of an exponential function is called a logarithmic function. Definition: x = a y if and only if y = log ax

48. Changing Between Logarithmic and Exponential Form

49. Logarithmic Functions • log 4 16 = 2 ↔ 42 = 16 • log 3 81 = 4 ↔ 34 = 81 • log10 100 = 2 ↔ 102 = 100

50. Inverses of Exponential Functions