College Algebra Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

1. College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson

2. Exponential and Logarithmic Functions 5

3. Chapter Overview • In this chapter, we study a new class of functions called exponential functions. • For example, f(x) = 2xis an exponential function (with base 2).

4. Chapter Overview • Notice how quickly the values of this function increase:f(3) = 23 = 8 f(10) = 210 = 1,024 f(30) = 230 = 1,073,741,824

5. Chapter Overview • Compare that with the function g(x) = x2where g(30) = 302 = 900. • The point is, when the variable is in the exponent, even a small change in the variable can cause a dramatic change in the value of the function.

6. Chapter Overview • In spite of this incomprehensibly huge growth, exponential functions are appropriate for modeling population growth for all living things—from bacteria to elephants.

7. Chapter Overview • To understand how a population grows, consider the case of a single bacterium, which divides every hour.

8. Chapter Overview • After one hour, we would have 2 bacteria; after two hours, 22 or 4 bacteria; after three hours, 23 or 8 bacteria; and so on. • After x hours, we would have 2xbacteria.

9. Chapter Overview • This leads us to model the bacteria population by the function f(x) = 2x

10. Chapter Overview • The principle governing population growth is: • The larger the population, the greater the number of offspring.

11. Chapter Overview • This same principle is present in many other real-life situations. • For example, the larger your bank account, the more interest you get. • So, we also use exponential functions to find compound interest.

12. Exponential Functions 5.1

13. Introduction • We now study one of the most important functions in mathematics—the exponential function. • This function is used to model such natural processes as population growth and radioactive decay.

14. Exponential Functions

15. Exponential Functions • In Section P.5, we defined axfor a > 0 and x a rational number. • However, we have not yet defined irrational powers. • So, what is meant by or 2π?

16. Exponential Functions • To define axwhen x is irrational, we approximate x by rational numbers. • For example, since is an irrational number, we successively approximate by these rational powers:

17. Exponential Functions • Intuitively, we can see that these rational powers of a are getting closer and closer to . • It can be shown using advanced mathematics that there is exactly one number that these powers approach. • We define to be this number.

18. Exponential Functions • For example, using a calculator, we find: • The more decimal places of we use in our calculation, the better our approximation of . • It can be proved that the Laws of Exponents are still true when the exponents are real numbers.

19. Exponential Function—Definition • The exponential function with base ais defined for all real numbers x by: f(x) = axwhere a > 0 and a≠ 1. • We assume a ≠ 1 because the function f(x) = 1x = 1 is just a constant function.

20. Exponential Functions • Here are some examples: f(x) = 2xg(x) = 3xh(x) = 10x

21. E.g. 1—Evaluating Exponential Functions • Let f(x) = 3x and evaluate the following: • f(2) • f(–⅔) • f(π) • f( ) • We use a calculator to obtain the values of f.

22. Example (a) E.g. 1—Evaluating Exp. Functions • Calculator keystrokes: 3, ^, 2, ENTER • Output: 9 • Thus, f(2) = 32 = 9

23. Example (b) E.g. 1—Evaluating Exp. Functions • Calculator keystrokes: 3, ^, (, (–), 2, ÷, 3, ), ENTER • Output: 0.4807498 • Thus, f(–⅔) = 3–⅔≈ 0.4807

24. Example (c) E.g. 1—Evaluating Exp. Functions • Calculator keystrokes: 3, ^, π, ENTER • Output: 31.5442807 • Thus, f(π) = 3π≈ 31.544

25. Example (d) E.g. 1—Evaluating Exp. Functions • Calculator keystrokes: 3, ^, √, 2, ENTER • Output: 4.7288043 • Thus, f( ) = ≈ 4.7288

26. Graphs of Exponential Functions

27. Graphs of Exponential Functions • We first graph exponential functions by plotting points. • We will see that these graphs have an easily recognizable shape.

28. E.g. 2—Graphing Exp. Functions by Plotting Points • Draw the graph of each function. • f(x) = 3x • g(x) = (⅓)x

29. E.g. 2—Graphing Exp. Functions by Plotting Points • First, we calculate values of f(x) and g(x).

30. E.g. 2—Graphing Exp. Functions by Plotting Points • Then, we the plot points to sketch the graphs.

31. E.g. 2—Graphing Exp. Functions by Plotting Points • Notice that: • So, we could have obtained the graph of gfrom the graph of fby reflecting in the y-axis.

32. Graphs of Exponential Functions • The figure shows the graphs of the family of exponential functions f(x) = ax for various values of the base a. • All these graphs pass through the point (0, 1) because a0 = 1 for a ≠ 0.

33. Graphs of Exponential Functions • You can see from the figure that there are two kinds of exponential functions: • If 0 < a < 1, the function decreases rapidly. • If a > 1, the function increases rapidly.

34. Graphs of Exponential Functions • The x-axis is a horizontal asymptote for the exponential function f(x) = ax. • This is because: • When a > 1, we have ax→ 0 as x → –∞. • When 0 < a < 1, we have ax→ 0 as x →∞.

35. Graphs of Exponential Functions • Also, ax> 0 for all . • So, the function f(x) = ax has domain and range (0, ∞). • These observations are summarized as follows.

36. Graphs of Exponential Functions • The exponential function f(x) = ax (a > 0, a≠ 1)has domain and range (0, ∞). • The line y = 0 (the x-axis) is a horizontal asymptote of f.

37. Graphs of Exponential Functions • The graph of f has one of these shapes.

38. E.g. 3—Identifying Graphs of Exponential Functions • Find the exponential function f(x) = ax whose graph is given.

39. Example (a) E.g. 3—Identifying Graphs • Since f(2) = a2 = 25, we see that the base is a = 5. • Thus, f(x) = 5x

40. Example (b) E.g. 3—Identifying Graphs • Since f(3) = a3 = 1/8 , we see that the base is a = ½ . • Thus, f(x) = (½)x

41. Graphs of Exponential Functions • In the next example, we see how to graph certain functions—not by plotting points—but by: • Taking the basic graphs of the exponential functions in Figure 2. • Applying the shifting and reflecting transformations of Section 3.5.

42. E.g. 4—Transformations of Exponential Functions • Use the graph of f(x) = 2x to sketch the graph of each function. • g(x) = 1 + 2x • h(x) = –2x • k(x) = 2x –1

43. Example (a) E.g. 4—Transformations • To obtain the graph of g(x) = 1 + 2x, we start with the graph of f(x) = 2x and shift it upward 1 unit. • Notice that the line y = 1 is now a horizontal asymptote.

44. Example (b) E.g. 4—Transformations • Again, we start with the graph of f(x) = 2x. • However, here, we reflect in the x-axis to get the graph of h(x) = –2x.

45. Example (c) E.g. 4—Transformations • This time, we start with the graph of f(x) = 2x and shift it to the right by 1 unit—to get the graph of k(x) = 2x–1.

46. E.g. 5—Comparing Exponential and Power Functions • Compare the rates of growth of the exponential function f(x) = 2x and the power function g(x) = x2 by drawing the graphs of both functions in these viewing rectangles. • [0, 3] by [0, 8] • [0, 6] by [ 0, 25] • [0, 20] by [0, 1000]

47. Example (a) E.g. 5—Exp. and Power Functions • The figure shows that the graph of g(x) = x2 catches up with, and becomes higher than, the graph of f(x) = 2x at x = 2.

48. Example (b) E.g. 5—Exp. and Power Functions • The larger viewing rectangle here shows that the graph of f(x) = 2x overtakes that of g(x) = x2 when x = 4.

49. Example (c) E.g. 5—Exp. and Power Functions • This figure gives a more global view and shows that, when x is large, f(x) = 2xis much larger than g(x) = x2.

50. The Natural Exponential Function