467 4.3 Logarithmic Functions Calculator graphs of logarithmic functions sometimes do not give an accurate picture of the behavior of the graphs near the vertical asymptotes. While it may seem as if the graph has an endpoint, this is not the case. The resolution of the calculator screen is not precise enough to indicate that the graph approaches the vertical asymptote as the value of x gets closer to it. Do not draw incorrect conclusions just because the calculator does not show this behavior. 7 The graphs in Figures 26 and 27 and the information with them suggest the following generalizations about the graphs of logarithmic functions of the form ƒ1x2 = loga x. Characteristics of the Graph of f 1x2 =loga x 1. The points A1 a , -1B, 11, 02, and 1a, 12 are on the graph. 2. If a 71, then ƒ is an increasing function. If 0 6a 61, then ƒ is a decreasing function. 3. The y-axis is a vertical asymptote. 4. The domain is 10, ∞2, and the range is 1-∞, ∞2. (b) Another way to graph a logarithmic function is to write ƒ1x2 =y =log3 x in exponential form as x =3 y. Then select y-values and calculate corresponding x-values. Several selected ordered pairs are shown in the table for the graph in Figure 29. S Now Try Exercise 49. EXAMPLE 3 Graphing Logarithmic Functions Graph each function. (a) ƒ1x2 = log1/2 x (b) ƒ1x2 = log3 x SOLUTION (a) One approach is to first graph y = A1 2B x , which defines the inverse function of ƒ, by plotting points. Some ordered pairs are given in the table with the graph shown in red in Figure 28. The graph of ƒ1x2 = log1/2 x is the reflection of the graph of y = A 1 2B x across the line y = x. The ordered pairs for y = log1/2 x are found by interchanging the x- and y-values in the ordered pairs for y = A1 2B x . See the graph in blue in Figure 28. x y –2 4 4 y = ( )x f(x) = log1/2 x 0 y = x 1 2 Figure 28 x y =A1 2Bx -2 4 -1 2 0 1 1 1 2 2 1 4 4 1 16 x ƒ1x2 =log1/2 x 4 -2 2 -1 1 0 1 2 1 1 4 2 1 16 4 3 9 –2 3 x y 0 f(x) = log3 x Figure 29 x ƒ1x2 =log3 x 1 31 1 0 3 1 9 2 Think: x = 3y
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