316 CHAPTER 5 Exponential and Logarithmic Functions Properties of the Logarithmic Function ( ) = > ≠ f x x a a a log ; 0, 1 • The domain is the set of positive real numbers, or ( )∞ 0, using interval notation; the range is the set of all real numbers, or ( ) −∞ ∞, using interval notation. • The x -intercept of the graph is 1. There is no y -intercept. • The y -axis ( ) = x 0 is a vertical asymptote of the graph of f. • A logarithmic function is decreasing if < < a 0 1 and is increasing if > a 1. • The graph of f contains the points ( ) 1, 0 , ( ) a, 1 , and ( ) − a 1 , 1. • The graph is smooth and continuous, with no corners or gaps. In Words = y x loge is written = y x ln . The graphs of = y x loga in Figures 35(a) and (b) on the previous page lead to the following properties. If the base of a logarithmic function is the number e , the result is the natural logarithm function . This function occurs so frequently in applications that it is given a special symbol, ln (from the Latin, logarithmus naturalis ). That is, = = y x x e ln if and only if y (1) Because = y x ln and the exponential function = y ex are inverse functions, the graph of = y x ln can be obtained by reflecting the graph of = y ex about the line = y x. See Figure 38. Using a TI-84 Plus CE with the ln key, we can obtain other points on the graph of ( ) = f x x ln . See Table 7. Figure 38 ( ) ( 21, x y5x y5ex y5In x y 5 0 x 5 0 y 21 4 23 3 0, 1 ) 1, 0 (1, e) (e, 1) ) ( 1 –e ( ) ,21 1 –e 3 Y 3 5 x Y 1 5 ex 22 24 4 Y2 5 In x Table 7 Graphing a Logarithmic Function and Its Inverse (a) Find the domain of the logarithmic function ( ) ( ) = − − f x x ln 2 . (b) Graph f. (c) From the graph, determine the range and vertical asymptote of f. (d) Find −f ,1 the inverse of f. (e) Find the domain and the range of −f .1 (f) Graph −f .1 Solution EXAMPLE 6 (a) The domain of f consists of all x for which − > x 2 0, or equivalently, > x 2. The domain of f is { } > x x 2 , or ( )∞ 2, in interval notation.
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