472 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Recall that for inverse functions ƒ and g, 1ƒ∘ g21x2 = 1g∘ ƒ21x2 = x. We can use this property with exponential and logarithmic functions to state two more properties. If ƒ1x2 = ax and g1x2 = log a x, then 1ƒ∘ g21x2 = aloga x = x and 1g∘ ƒ21x2 = log a 1ax2 = x. Theorem on Inverses For a 70, a≠1, the following properties hold true. alog a x =x 1for x +02 and log a ax =x For example, 7log7 10 = 10, log 5 53 = 3, and log r r k+1 = k + 1. The second statement in the theorem will be useful when we solve logarithmic and exponential equations. 4.3 Exercises CONCEPT PREVIEW Match the logarithm in Column I with its value in Column II. Remember that loga x is the exponent to which a must be raised in order to obtain x. I 1. (a) log2 16 (b) log3 1 (c) log10 0.1 (d) log2 22 (e) loge 1 e2 (f) log1/2 8 II A. 0 B. 1 2 C. 4 D. -3 E. -1 F. -2 I 2. (a) log3 81 (b) log3 1 3 (c) log10 0.01 (d) log6 26 (e) loge 1 (f) log3 273/2 II A. -2 B. -1 C. 0 D. 1 2 E. 9 2 F. 4 CONCEPT PREVIEW Write each equivalent form. 3. Write log2 8 = 3 in exponential form. 4. Write 103 = 1000 in logarithmic form. CONCEPT PREVIEW Solve each logarithmic equation. 5. logx 16 81 = 2 6. log36 2 3 6 = x CONCEPT PREVIEW Sketch the graph of each function. Give the domain and range. 7. ƒ1x2 = log5 x 8. g1x2 = log1/5 x NOTE The values in Example 7 are approximations of logarithms, so the final digit may differ from the actual 4-decimal-place approximation after properties of logarithms are applied.
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