1. 4.3 - 1 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 4 Inverse, Exponential, and Logarithmic Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc.

2. 2 4.3 Logarithmic Functions Logarithms Logarithmic Equations Logarithmic Functions Properties of Logarithms

3. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Logarithms The previous section dealt with exponential functions of the form y = a x for all positive values of a, where a ≠ 1. The horizontal line test shows that exponential functions are one-to-one, and thus have inverse functions.

4. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 Logarithms The equation defining the inverse of a function is found by interchanging x and y in the equation that defines the function. Starting with y = a x and interchanging x and y yields

5. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 Logarithms Here y is the exponent to which a must be raised in order to obtain x. We call this exponent a logarithm, symbolized by the abbreviation “log.” The expression log a x represents the logarithm in this discussion. The number a is called the base of the logarithm, and x is called the argument of the expression. It is read “logarithm with base a of x,” or “logarithm of x with base a.”

6. 4.3 - 6 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 Logarithm For all real numbers y and all positive numbers a and x, where a ≠ 1, if and only if The expression log a x represents the exponent to which the base a must be raised in order to obtain x.

7. 4.3 - 7 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 Example 1 WRITING EQUIVALENT LOGARITHMIC AND EXPONENTIAL FORMS The table shows several pairs of equivalent statements, written in both logarithmic and exponential forms.

8. 4.3 - 8 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 Logarithmic form: y = log a x Exponent Base Exponential form: a y = x Exponent Base Example 1 WRITING EQUIVALENT LOGARITHMIC AND EXPONENTIAL FORMS To remember the relationships among a, x, and y in the two equivalent forms y = log a x and x = a y, refer to these diagrams.

9. 4.3 - 9 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 Example 2 SOLVING LOGARITHMIC EQUATIONS Solve each equation. Solution (a) Write in exponential form. Take cube roots

10. 4.3 - 10 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Example 2 SOLVING LOGARITHMIC EQUATIONS Check Let x = 2/3. Write in exponential form. Original equation True The solution set is

11. 4.3 - 11 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 Example 2 SOLVING LOGARITHMIC EQUATIONS Solve each equation. Solution (b) Write in exponential form. The solution set is {32}. Apply the exponent.

12. 4.3 - 12 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 Example 2 SOLVING LOGARITHMIC EQUATIONS Solve each equation. Solution (c) Write in exponential form. The solution set is Write with the same base. Power rule for exponents. Set exponents equal. Divide by 2.

13. 4.3 - 13 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 Logarithmic Function If a > 0, a ≠ 1, and x > 0, then defines the logarithmic function with base a.

14. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 Logarithmic Functions Exponential and logarithmic functions are inverses of each other. The graph of y = 2 x is shown in red. The graph of its inverse is found by reflecting the graph of y = 2 x across the line y = x.

15. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 Logarithmic Functions The graph of the inverse function, defined by y = log 2 x, shown in blue, has the y-axis as a vertical asymptote.

16. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 Logarithmic Functions Since the domain of an exponential function is the set of all real numbers, the range of a logarithmic function also will be the set of all real numbers. In the same way, both the range of an exponential function and the domain of a logarithmic function are the set of all positive real numbers. Thus, logarithms can be found for positive numbers only.

17. 4.3 - 17 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 Domain: (0,  ) Range: (– ,  ) LOGARITHMIC FUNCTION x (x)(x) ¼ – 2 ½– 1– 1 10 21 42 83   (x) = log a x, for a > 1, is increasing and continuous on its entire domain, (0,  ). For  (x) = log 2 x:

18. 4.3 - 18 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 Domain: (0,  ) Range: (– ,  ) LOGARITHMIC FUNCTION x (x)(x) ¼ – 2 ½– 1– 1 10 21 42 83  The y-axis is a vertical asymptote as x  0 from the right. For  (x) = log 2 x:

19. 4.3 - 19 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19 Domain: (0,  ) Range: (– ,  ) LOGARITHMIC FUNCTION x (x)(x) ¼ – 2 ½– 1– 1 10 21 42 83  The graph passes through the points For  (x) = log 2 x:

20. 4.3 - 20 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20 Domain: (0,  ) Range: (– ,  ) LOGARITHMIC FUNCTION x (x)(x) ¼ 2 ½1 10 2 – 1– 1 4 – 2– 2 8 – 3– 3   (x) = log a x, for 0 < a < 1, is decreasing and continuous on its entire domain, (0,  ). For  (x) = log 1/2 x:

21. 4.3 - 21 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21 Domain: (0,  ) Range: (– ,  ) LOGARITHMIC FUNCTION x (x)(x) ¼ 2 ½1 10 2 – 1– 1 4 – 2– 2 8 – 3– 3 For  (x) = log 1/2 x:  The y-axis is a vertical asymptote as x  0 from the right.

22. 4.3 - 22 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22 Domain: (0,  ) Range: (– ,  ) LOGARITHMIC FUNCTION x (x)(x) ¼ 2 ½1 10 2 – 1– 1 4 – 2– 2 8 – 3– 3 For  (x) = log 1/2 x:  The graph passes through the points

23. 4.3 - 23 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23 Characteristics of the Graph of 1. The points are on the graph. 2. If a > 1, then  is an increasing function. If 0 < a < 1, then  is a decreasing function. 3. The y-axis is a vertical asymptote. 4. The domain is (0,  ), and the range is (– ,  ).

24. 4.3 - 24 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24 Example 3 GRAPHING LOGARITHMIC FUNCTIONS Graph each function. Solution (a) First graph y = (½) x which defines the inverse function of , by plotting points. The graph of  (x) = log 1/2 x is the reflection of the graph of y = (½) x across the line y = x. The ordered pairs for y = log 1/2 x are found by interchanging the x- and y-values in the ordered pairs for y = (½) x.

25. 4.3 - 25 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 25 Example 3 GRAPHING LOGARITHMIC FUNCTIONS Solution Graph each function. (a)

26. 4.3 - 26 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 26 Example 3 GRAPHING LOGARITHMIC FUNCTIONS Solution (b) Another way to graph a logarithmic function is to write  (x) = y = log 3 x in exponential form as x = 3 y, and then select y-values and calculate corresponding x-values. Graph each function.

27. 4.3 - 27 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27 Example 3 GRAPHING LOGARITHMIC FUNCTIONS Solution Graph each function. (b)

28. 4.3 - 28 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 28 Caution If you write a logarithmic function in exponential form to graph, as in Example 3(b), start first with y-values to calculate corresponding x-values. Be careful to write the values in the ordered pairs in the correct order.

29. 4.3 - 29 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29 Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution (a) The graph of  (x) = log 2 (x – 1) is the graph of  (x) = log 2 x translated 1 unit to the right. The vertical asymptote has equation x = 1. Since logarithms can be found only for positive numbers, we solve x – 1 > 0 to find the domain, (1,  ). To determine ordered pairs to plot, use the equivalent exponential form of the equation y = log 2 (x – 1).

30. 4.3 - 30 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30 Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution Write in exponential form. Add 1. (a)

31. 4.3 - 31 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 31 Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution We first choose values for y and then calculate each of the corresponding x-values. The range is (– ,  ). (a)

32. 4.3 - 32 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32 Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution (b) The function  (x) = (log 3 x) – 1 has the same graph as g (x) = log 3 x translated 1 unit down. We find ordered pairs to plot by writing the equation y = (log 3 x) – 1 in exponential form.

33. 4.3 - 33 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 33 Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution Add 1. Write in exponential form. (b)

34. 4.3 - 34 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 34 Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution Again, choose y-values and calculate the corresponding x-values. The domain is (0,  ) and the range is (– ,  ). (b)

35. 4.3 - 35 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 35 Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution (c) The graph of f (x) = log 4 (x + 2) + 1 is obtained by shifting the graph of y = log 4 x to the left 2 units and up 1 unit. The domain is found by solving x + 2 > 0, which yields (–2,  ). The vertical asymptote has been shifted to the left 2 units as well, and it has equation x = –2. The range is unaffected by the vertical shift and remains (– ,  ).

36. 4.3 - 36 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 36 Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Graph each function. Give the domain and range. Solution (c)

37. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 37 Properties of Logarithms The properties of logarithms enable us to change the form of logarithmic statements so that products can be converted to sums, quotients can be converted to differences, and powers can be converted to products.

38. 4.3 - 38 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 38 Properties of Logarithms For x > 0, y > 0, a > 0, a ≠ 1, and any real number r, the following properties hold. Description The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers. Property Product Property

39. 4.3 - 39 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 39 Properties of Logarithms For x > 0, y > 0, a > 0, a ≠ 1, and any real number r, the following properties hold. Quotient Property Description The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers. Property

40. 4.3 - 40 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 40 Properties of Logarithms Power Property Description The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number. For x > 0, y > 0, a > 0, a ≠ 1, and any real number r, the following properties hold. Property

41. 4.3 - 41 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 41 Properties of Logarithms Logarithm of 1 Description The base a logarithm of 1 is 0. The base a logarithm of a is 1. For x > 0, y > 0, a > 0, a ≠ 1, and any real number r, the following properties hold. Property Base a Logarithm of a

42. 4.3 - 42 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 42 Example 5 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution (a) Product property

43. 4.3 - 43 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 43 Example 5 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution (b) Quotient property

44. 4.3 - 44 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 44 Example 5 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution (c) Power property

45. 4.3 - 45 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 45 Example 5 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution (d) Use parentheses to avoid errors. Be careful with signs.

46. 4.3 - 46 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 46 Example 5 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution (e)

47. 4.3 - 47 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 47 Example 5 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Power property Product and quotient properties (f) Solution

48. 4.3 - 48 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 48 Example 5 USING THE PROPERTIES OF LOGARITHMS Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution (f) Power property Distributive property

49. 4.3 - 49 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 49 Example 6 USING THE PROPERTIES OF LOGARITHMS Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution (a) Product and quotient properties

50. 4.3 - 50 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 50 Example 6 USING THE PROPERTIES OF LOGARITHMS Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution (b) Power property Quotient property

51. 4.3 - 51 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 51 Example 6 USING THE PROPERTIES OF LOGARITHMS Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution (c) Power property

52. 4.3 - 52 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 52 Example 6 USING THE PROPERTIES OF LOGARITHMS Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution Product and quotient properties Rules for exponents (c)

53. 4.3 - 53 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 53 Example 6 USING THE PROPERTIES OF LOGARITHMS Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. Solution Rules for exponents Definition of a 1/n (c)

54. 4.3 - 54 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 54 Caution There is no property of logarithms to rewrite a logarithm of a sum or difference. That is why, in Example 6(a), log 3 (x + 2) was not written as log 3 x + log 3 2. The distributive property does not apply in a situation like this because log 3 (x + y) is one term. The abbreviation “log” is a function name, not a factor.

55. 4.3 - 55 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 55 Example 7 USING THE PROPERTIES OF LOGARITHMS WITH NUMERICAL VALUES Assume that log 10 2 = 0.3010. Find each logarithm. Solution (a)

56. 4.3 - 56 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 56 Example 7 USING THE PROPERTIES OF LOGARITHMS WITH NUMERICAL VALUES Assume that log 10 2 = 0.3010. Find each logarithm. Solution (b)

57. 4.3 - 57 Copyright © 2013, 2009, 2005 Pearson Education, Inc. 57 Theorem on Inverses For a > 0, a ≠ 1, the following properties hold.

58. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 58 Theorem on Inverses The following are examples of applications of this theorem. The second statement in the theorem will be useful in Sections 4.5 and 4.6 when we solve other logarithmic and exponential equations.