466 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions The domain of an exponential function is the set of all real numbers, so the range of a logarithmic function also will be the set of all real numbers. In the same way, both the range of an exponential function and the domain of a logarithmic function are the set of all positive real numbers. Thus, logarithms can be found for positive numbers only. Logarithmic Function f 1x2 =loga x Domain: 10, ∞2 Range: 1-∞, ∞2 For ƒ1x2 = log2 x: x ƒ1x2 1 42 1 21 1 0 2 1 4 2 8 3 • ƒ1x2 =log a x, for a +1, is increasing and continuous on its entire domain, 10, ∞2. • The y-axis is a vertical asymptote as x S0 from the right. • The graph passes through the points A1 a , -1B, 11, 02, and 1a, 12. For ƒ1x2 = log1/2 x: x ƒ1x2 1 4 2 1 2 1 1 0 2 -1 4 -2 8 -3 • ƒ1x2 =log a x, for 0 *a *1, is decreasing and continuous on its entire domain, 10, ∞2. • The y-axis is a vertical asymptote as x S0 from the right. • The graph passes through the points A1 a , -1B, 11, 02, and 1a, 12. f(x) = loga x, a > 1 x y 0 (1, 0) (a, 1) ( , –1) a 1 f(x) = loga x, a > 1 Figure 26 This is the general behavior seen on a calculator graph for any base a, for a +1. f(x) = loga x, 0 < a < 1 x y 0 (1, 0) (a, 1) ( , –1) a 1 f(x) = loga x, 0 < a < 1 Figure 27 This is the general behavior seen on a calculator graph for any base a, for 0 *a *1.
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