SECTION 5.4 Logarithmic Functions 319 Figure 43 f 21(x) 5 10 x/3 1 1 (2, 0) (0, 2) x y 12 10 12 22 4 6 8 10 22 4 6 8 y 5 x x 5 1 y 5 1 f (x) 5 3 log (x 2 1) (11, 3) (3, 11) (e) The domain of −f 1 is the range of f, which is the set of all real numbers, from part (c).The range of −f 1 is the domain of f, which is ( )∞ 1, in interval notation. (f) To graph −f ,1 use the graph of f in Figure 42(c) and reflect it about the line = y x. See Figure 43. We could also graph ( ) = + −f x 10 1 x 1 3 using transformations. Now Work PROBLEM 83 5 Solve Logarithmic Equations Equations that contain logarithms are called logarithmic equations. Care must be taken when solving logarithmic equations algebraically. In the expression M log , a remember that a and M are positive and ≠ a 1. Be sure to check each apparent solution in the original equation and discard any that are extraneous. Some logarithmic equations can be solved by changing the logarithmic equation to exponential form using the fact that = y x loga means = a x. y Solving Logarithmic Equations Solve: (a) ( ) − = x log 4 7 2 3 (b) = log 64 2 x Solution EXAMPLE 8 (a) To solve, change the logarithmic equation to exponential form. ( ) − = − = − = = = x x x x x log 4 7 2 4 7 3 4 7 9 4 16 4 3 2 Change to exponential form. Check: ( ) ( ) − = ⋅ − = = x log 4 7 log 44 7 log9 2 3 3 3 = 3 9 2 The solution set is { }4 . (b) To solve, change the logarithmic equation to exponential form. = = = ± = ± x x log 64 2 64 64 8 x 2 Change to exponential form. Use the Square Root Method. Recall = x ay means = y x log a and that > ≠ a a 0, 1. The base of a logarithm must be positive, so discard −8. Check the potential solution 8. Check: = log 64 2 8 = 8 64 2 The solution set is { }8 . Using Logarithms to Solve an Exponential Equation Solve: = e 5 x2 Solution EXAMPLE 9 To solve, change the exponential equation to logarithmic form. = = = ≈ e x x 5 ln5 2 ln5 2 0.805 x2 Change to logarithmic form. Exact solution Approximate solution The solution set is { } ln5 2 . Now Work PROBLEMS 91 AND 103
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