492 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Logarithmic Equations The following equations involve logarithms of variable expressions. EXAMPLE 5 Solving Logarithmic Equations Solve each equation. Give exact values. (a) 7 ln x = 28 (b) log2 1x 3 - 192 = 3 SOLUTION (a) 7 ln x = 28 loge x = 4 ln x = loge x; Divide by 7. x = e4 Write in exponential form. The solution set is 5e46. (b) log2 1x 3 - 192 = 3 x3 - 19 = 23 Write in exponential form. x3 - 19 = 8 Apply the exponent. x3 = 27 Add 19. x = 23 27 Take cube roots. x = 3 23 27 = 3 The solution set is 536. S Now Try Exercises 41 and 49. EXAMPLE 6 Solving a Logarithmic Equation Solve log 1x + 62 - log 1x + 22 = log x. Give exact value(s). SOLUTION Recall that logarithms are defined only for nonnegative numbers. log 1x + 62 - log 1x + 22 = log x log x + 6 x + 2 = log x Quotient property x + 6 x + 2 = x Property of logarithms x + 6 = x1x + 22 Multiply by x + 2. x + 6 = x2 + 2x Distributive property x2 + x - 6 = 0 Write in standard form. 1x + 321x - 22 = 0 Factor. x + 3 = 0 or x - 2 = 0 Zero-factor property x = -3 or x = 2 Solve for x. The proposed negative solution 1-32 is not in the domain of log x in the original equation because it leads to the logarithm of a negative number. The only valid solution is the positive number 2. The solution set is 526. S Now Try Exercise 69.
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