464 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions EXAMPLE 2 Solving Logarithmic Equations Solve each equation. (a) logx 8 27 = 3 (b) log4 x = 5 2 (c) log492 3 7 = x SOLUTION Many logarithmic equations can be solved by first writing the equation in exponential form. (a) logx 8 27 = 3 x3 = 8 27 Write in exponential form. x3 = a 2 3b 3 8 27 = A 2 3B 3 x = 2 3 Take cube roots. CHECK logx 8 27 = 3 Original equation log2/3 8 27 ≟3 Let x = 2 3. a2 3b 3 ≟ 8 27 Write in exponential form. 8 27 = 8 27 ✓ True The solution set is E2 3F. EXAMPLE 1 Writing Equivalent Logarithmic and Exponential Forms The table shows several pairs of equivalent statements, written in both logarithmic and exponential forms. SOLUTION Logarithmic Form Exponential Form log2 8 = 3 23 = 8 log1/2 16 = -4 A 1 2B -4 = 16 log10 100,000 = 5 105 = 100,000 log3 1 81 = -4 3-4 = 1 81 log5 5 = 1 51 = 5 log3/4 1 = 0 A 3 4B 0 = 1 To remember the relationships among a, x, and y in the two equivalent forms y = loga x and x = ay, refer to these diagrams. A logarithm is an exponent. Exponent Logarithmic form: y = loga x Base Exponent Exponential form: ay = x Base S Now Try Exercises 11, 13, 15, and 17. Logarithmic Equations The definition of logarithm can be used to solve a logarithmic equation, which is an equation with a logarithm in at least one term.
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