1. Copyright © Cengage Learning. All rights reserved. 11 Exponential and Logarithmic Functions

2. 11.1 EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Copyright © Cengage Learning. All rights reserved.

3. 3 Recognize and evaluate exponential functions with base a. Graph exponential functions and use the One-to-One Property. Recognize, evaluate, and graph exponential functions with base e. Use exponential functions to model and solve real-life problems. What You Should Learn

4. 4 Exponential Functions

5. 5 So far, this text has dealt mainly with algebraic functions, which include polynomial functions and rational functions. In this chapter, you will study two types of nonalgebraic functions–exponential functions and logarithmic functions. These functions are examples of transcendental functions.

6. 6 Exponential Functions The base a = 1 is excluded because it yields f (x) = 1 x = 1. This is a constant function, not an exponential function. You have evaluated a x for integer and rational values of x. For example, you know that 4 3 = 64 and 4 1/2 = 2. However, to evaluate 4 x for any real number x, you need to interpret forms with irrational exponents.

7. 7 Exponential Functions For the purposes of this text, it is sufficient to think of (where  1.41421356) as the number that has the successively closer approximations a 1.4, a 1.41, a 1.414, a 1.4142, a 1.41421,....

8. 8 Example 1 – Evaluating Exponential Functions Use a calculator to evaluate each function at the indicated value of x. Function Value a. f (x) = 2 x x = –3.1 b. f (x) = 2 –x x =  c. f (x) = 0.6 x x =

9. 9 Example 1 – Solution Function Value Graphing Calculator Keystrokes Display a. f (–3.1) = 2 –3.1 0.1166291 b. f (  ) = 2 –  0.1133147 c. f = 0.6 3/2 0.4647580

10. 10 Graphs of Exponential Functions

11. 11 Example 2 – Graphs of y = a x In the same coordinate plane, sketch the graph of each function. a. f (x) = 2 x b. g(x) = 4 x

12. 12 Example 2 – Solution The table below lists some values for each function, and Figure 3.1 shows the graphs of the two functions. Note that both graphs are increasing. Moreover, the graph of g(x) = 4 x is increasing more rapidly than the graph of f (x) = 2 x. Figure 3.1

13. 13 Graphs of Exponential Functions The basic characteristics of exponential functions y = a x and y = a –x are summarized in Figures 3.3 and 3.4. Graph of y = a x, a > 1 Domain: (, ) Range: (0, ) y-intercept: (0, 1) Increasing x-axis is a horizontal asymptote (a x → 0, as x→ ). Continuous Figure 3.3

14. 14 Graphs of Exponential Functions Graph of y = a –x, a > 1 Domain: (, ) Range: (0, ) y-intercept: (0, 1) Decreasing x-axis is a horizontal asymptote (a –x → 0, as x→ ). Continuous From Figures 3.3 and 3.4, you can see that the graph of an exponential function is always increasing or always decreasing. Figure 3.4

15. 15 Graphs of Exponential Functions As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one-to-one functions. You can use the following One-to-One Property to solve simple exponential equations. For a > 0 and a ≠ 1, a x = a y if and only if x = y. One-to-One Property

16. 16 The Natural Base e

17. 17 The Natural Base e In many applications, the most convenient choice for a base is the irrational number e  2.718281828.... This number is called the natural base. The function given by f (x) = e x is called the natural exponential function. Its graph is shown in Figure 3.9. Figure 3.9

18. 18 The Natural Base e Be sure you see that for the exponential function f (x) = e x, e is the constant 2.718281828..., whereas x is the variable.

19. 19 Example 6 – Evaluating the Natural Exponential Function Use a calculator to evaluate the function given by f (x) = e x at each indicated value of x. a. x = –2 b. x = –1 c. x = 0.25 d. x = –0.3

20. 20 Example 6 – Solution Function Value Graphing Calculator Keystrokes Display a. f (–2) = e –2 0.1353353 b. f (–1) = e –1 0.3678794 c. f (0.25) = e 0.25 1.2840254 d. f (–0.3) = e –0.3 0.7408182

21. 21 Definition of the Exponential Function The exponential function f with base b is defined by f (x) = b x or y = b x Where b is a positive constant other than and x is any real number. The exponential function f with base b is defined by f (x) = b x or y = b x Where b is a positive constant other than and x is any real number. / Here are some examples of exponential functions. f (x) = 2 x g(x) = 10 x h(x) = 3 x+1 Base is 2.Base is 10.Base is 3.

22. 22 Text Example The exponential function f (x) = 13.49(0.967) x – 1 describes the number of O-rings expected to fail, when the temperature is x°F. On the morning the Challenger was launched, the temperature was 31°F, colder than any previous experience. Find the number of O-rings expected to fail at this temperature. SolutionBecause the temperature was 31°F, substitute 31 for x and evaluate the function at 31. f (x) = 13.49(0.967) x – 1 This is the given function. f (31) = 13.49(0.967) 31 – 1 Substitute 31 for x. f (31) = 13.49(0.967) 31 – 1=3.77

23. 23 Characteristics of Exponential Functions 1.The domain of f (x) = b x consists of all real numbers. The range of f (x) = b x consists of all positive real numbers. 2.The graphs of all exponential functions pass through the point (0, 1) because f (0) = b 0 = 1. 3.If b > 1, f (x) = b x has a graph that goes up to the right and is an increasing function. 4.If 0 < b < 1, f (x) = b x has a graph that goes down to the right and is a decreasing function. 5.f (x) = b x is a one-to-one function and has an inverse that is a function. 6.The graph of f (x) = b x approaches but does not cross the x-axis. The x-axis is a horizontal asymptote. 1.The domain of f (x) = b x consists of all real numbers. The range of f (x) = b x consists of all positive real numbers. 2.The graphs of all exponential functions pass through the point (0, 1) because f (0) = b 0 = 1. 3.If b > 1, f (x) = b x has a graph that goes up to the right and is an increasing function. 4.If 0 < b < 1, f (x) = b x has a graph that goes down to the right and is a decreasing function. 5.f (x) = b x is a one-to-one function and has an inverse that is a function. 6.The graph of f (x) = b x approaches but does not cross the x-axis. The x-axis is a horizontal asymptote. f (x) = b x b > 1 f (x) = b x 0 < b < 1

24. 24 Transformations Involving Exponential Functions Shifts the graph of f (x) = b x upward c units if c > 0. Shifts the graph of f (x) = b x downward c units if c < 0. g(x) = -b x + cVertical translation Reflects the graph of f (x) = b x about the x-axis. Reflects the graph of f (x) = b x about the y-axis. g(x) = -b x g(x) = b -x Reflecting Multiplying y-coordintates of f (x) = b x by c, Stretches the graph of f (x) = b x if c > 1. Shrinks the graph of f (x) = b x if 0 < c < 1. g(x) = c b x Vertical stretching or shrinking Shifts the graph of f (x) = b x to the left c units if c > 0. Shifts the graph of f (x) = b x to the right c units if c < 0. g(x) = b x+c Horizontal translation DescriptionEquationTransformation

25. 25 Text Example Use the graph of f (x) = 3 x to obtain the graph of g(x) = 3 x+1. SolutionExamine the table below. Note that the function g(x) = 3 x+1 has the general form g(x) = b x+c, where c = 1. Because c > 0, we graph g(x) = 3 x+1 by shifting the graph of f (x) = 3 x one unit to the left. We construct a table showing some of the coordinates for f and g to build their graphs. f (x) = 3 x g(x) = 3 x+1 (0, 1) (-1, 1) 123456 -5-4-3-2

26. 26 Problems Sketch a graph using transformation of the following: 1. 2. 3. Recall the order of shifting: horizontal, reflection (horz., vert.), vertical.

27. 27 The Natural Base e An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to 2.72. More accurately, The number e is called the natural base. The function f (x) = e x is called the natural exponential function. f (x) = e x f (x) = 2 x f (x) = 3 x (0, 1) (1, 2) 1 2 3 4 (1, e) (1, 3)

28. 28 Formulas for Compound Interest After t years, the balance, A, in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas: 1.For n compoundings per year: 2.For continuous compounding: A = Pe rt.

29. 29 Example:Choosing Between Investments You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment? SolutionThe better investment is the one with the greater balance in the account after 6 years. Let’s begin with the account with monthly compounding. We use the compound interest model with P = 8000, r = 7% = 0.07, n = 12 (monthly compounding, means 12 compoundings per year), and t = 6. The balance in this account after 6 years is $12,160.84. more

30. 30 Example:Choosing Between Investments You want to invest $8000 for 6 years, and you have a choice between two accounts. The first pays 7% per year, compounded monthly. The second pays 6.85% per year, compounded continuously. Which is the better investment? SolutionFor the second investment option, we use the model for continuous compounding with P = 8000, r = 6.85% = 0.0685, and t = 6. The balance in this account after 6 years is $12,066.60, slightly less than the previous amount. Thus, the better investment is the 7% monthly compounding option.

31. 31 Example Use A= Pe rt to solve the following problem: Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if the money is compounded continuously Solution: A= Pe rt A = 2000e (.07)(8) A = 2000 e (.56) A = 2000 * 1.75 A = $3500

32. 32 LOGARITMIC FUNCTION

33. 33 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 33 Definition: Logarithmic Function For x  0 and 0  a  1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = log a x is equivalent to x = a y A logarithmic function is the inverse function of an exponential function. Exponential function:y = a x Logarithmic function:y = log a x is equivalent to x = a y A logarithm is an exponent!

34. 34 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 34 Examples: Write Equivalent Equations y = log 2 ( ) = 2 y Examples: Write the equivalent exponential equation and solve for y. 1 = 5 y y = log 5 1 16 = 4 y y = log 4 16 16 = 2 y y = log 2 16 SolutionEquivalent Exponential Equation Logarithmic Equation 16 = 2 4  y = 4 = 2 -1  y = –1 16 = 4 2  y = 2 1 = 5 0  y = 0

35. 35 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 35 log 10 –4LOG –4 ENTERERROR no power of 10 gives a negative number Common Logarithmic Function The base 10 logarithm function f (x) = log 10 x is called the common logarithm function. The LOG key on a calculator is used to obtain common logarithms. Examples: Calculate the values using a calculator. log 10 100 log 10 5 Function ValueKeystrokesDisplay LOG 100 ENTER2 LOG 5 ENTER0.6989700 log 10 ( ) – 0.3979400LOG ( 2 5 ) ENTER

36. 36 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 36 Properties of Logarithms Examples: Solve for x: log 6 6 = x log 6 6 = 1 property 2  x = 1 Simplify: log 3 3 5 log 3 3 5 = 5 property 3 Simplify: 7 log 7 9 7 log 7 9 = 9 property 3 Properties of Logarithms 1. log a 1 = 0 since a 0 = 1. 2. log a a = 1 since a 1 = a. 4. If log a x = log a y, then x = y. one-to-one property 3. log a a x = x and a log a x = x inverse property

37. 37 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 37 x y Graph f(x) = log 2 x Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x. 83 42 21 10 –1 –2 2x2x x y = log 2 x y = x y = 2 x (1, 0) x-intercept horizontal asymptote y = 0 vertical asymptote x = 0

38. 38 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 38 Example: f(x) = log 0 x Example: Graph the common logarithm function f(x) = log 10 x. by calculator 10.6020.3010–1–2f(x) = log 10 x 10421x y x 5 –5 f(x) = log 10 x x = 0 vertical asymptote (0, 1) x-intercept

39. 39 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 39 Graphs of Logarithmic Functions The graphs of logarithmic functions are similar for different values of a. f(x) = log a x (a  1) 3. x-intercept (1, 0) 5. increasing 6. continuous 7. one-to-one 8. reflection of y = a x in y = x 1. domain 2. range 4. vertical asymptote Graph of f (x) = log a x (a  1) x y y = x y = log 2 x y = a x domain range y-axis vertical asymptote x-intercept (1, 0)

40. 40 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 40 Natural Logarithm ic Function The function defined by f(x) = log e x = ln x is called the natural logarithm function. Use a calculator to evaluate: ln 3, ln –2, ln 100 ln 3 ln –2 ln 100 Function ValueKeystrokesDisplay LN 3 ENTER1.0986122 ERRORLN –2 ENTER LN 100 ENTER4.6051701 y = ln x (x  0, e 2.718281  ) y x 5 –5 y = ln x is equivalent to e y = x

41. 41 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 41 Properties of Natural Logarithms 1. ln 1 = 0 since e 0 = 1. 2. ln e = 1 since e 1 = e. 3. ln e x = x and e ln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression. inverse property property 2 property 1

42. 42 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 42 Example: Carbon Dating Example: The formula (t in years) is used to estimate the age of organic material. The ratio of carbon 14 to carbon 12 in a piece of charcoal found at an archaeological dig is. How old is it? To the nearest thousand years the charcoal is 57,000 years old. original equation multiply both sides by 10 12 take the natural log of both sides inverse property