465 4.3 Logarithmic Functions (b) log4 x = 5 2 45/2 = x Write in exponential form. 141/225 = x amn = 1am2n 25 = x 41/2 = 12221/2 = 2 32 = x Apply the exponent. CHECK log4 32 ≟5 2 Let x = 32. 45/2 ≟32 25 ≟32 45/2 = A 24 B 5 = 25 32 = 32 ✓ True The solution set is 5326. (c) log492 3 7 = x 49 x = 23 7 Write in exponential form. 1722x = 71/3 Write with the same base. 72x = 71/3 Power rule for exponents 2x = 1 3 Set exponents equal. x = 1 6 Divide by 2. A check shows that the solution set is E1 6F. S Now Try Exercises 21, 33, and 35. Logarithmic Functions We define the logarithmic function with base a. Logarithmic Function If a 70, a≠1, and x 70, then the logarithmic function with base a is ƒ1x2 =log a x. Exponential and logarithmic functions are inverses of each other. To show this, we use the three steps for finding the inverse of a function. ƒ1x2 = 2x Exponential function with base 2 y = 2x Let y = ƒ1x2. Step 1 x = 2y Interchange x and y. Step 2 y = log2 x Solve for y by writing in equivalent logarithmic form. Step 3 ƒ-11x2 = log 2 x Replace y with ƒ-11x2. The graph of ƒ1x2 = 2x has the x-axis as horizontal asymptote and is shown in red in Figure 25. Its inverse, ƒ-11x2 = log 2 x, has the y-axis as vertical asymptote and is shown in blue. The graphs are reflections of each other across the line y = x. As a result, their domains and ranges are interchanged. x y –2 –2 6 4 8 4 6 8 f –1(x) = log 2 x Domain: (0, ∞) Range: (–∞, ∞) 0 y = x (0, 1) (1, 2) (2, 1) (1, 0) ( , –1) 1 2 (–1, ) 1 2 f(x) = 2x Domain: (–∞, ∞) Range: (0, ∞) Figure 25 x ƒ1x2 =2 x -2 1 4 -1 1 2 0 1 1 2 2 4 x ƒ−11x2 =log 2 x 1 4 -2 1 2 -1 1 0 2 1 4 2
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