318 CHAPTER 5 Exponential and Logarithmic Functions If the base of a logarithmic function is the number 10, the result is the common logarithm function . If the base a of the logarithmic function is not indicated, it is understood to be 10. That is, = = y x x log if and only if 10y Because = y x log and the exponential function = y 10x are inverse functions, the graph of = y x log can be obtained by reflecting the graph of = y 10x about the line = y x. See Figure 41. Figure 41 (0, 1) (1, 0) x y 22 4 4 22 y 5 log x y 5 x ( ) 21 ( ) , 21 y 5 10 x 1 –– 10 , 1 –– 10 Graphing a Logarithmic Function and Its Inverse (a) Find the domain of the logarithmic function ( ) ( ) = − f x x 3log 1 . (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 7 (a) The domain of f consists of all x for which − > x 1 0, or equivalently, > x 1. The domain of f is { } > x x 1 , or ( )∞ 1, in interval notation. (b) To obtain the graph of ( ) = − y x 3log 1 , begin with the graph of = y x log and use transformations. See Figure 42. Figure 42 2 x y 6 8 10 12 Replace x by x – 1; horizontal shift right 1 unit. Multiply by 3; vertical stretch by a factor of 3. 2 4 –2 –2 (10, 1) (1, 0) (2, 0) (2, 0) x 5 0 x 5 1 2 x y 2 4 6 8 10 12 –2 –2 (11, 1) x 5 1 (a) y 5 log x (b) y 5 log (x 2 1) (c) y 5 3 log (x 2 1) 2 x y 2 4 6 8 10 12 –2 –2 (11, 3) (11 10 ,21) ( 1 10 ,21) (11 10 ,23) (c) The range of ( ) ( ) = − f x x 3log 1 is the set of all real numbers. The vertical asymptote is = x 1. (d) Begin with ( ) = − y x 3log 1 . The inverse function is defined implicitly by the equation ( ) = − x y 3log 1 Solve for y . ( ) = − = − = + x y y y 3 log 1 10 1 10 1 x x 3 3 The inverse of f is ( ) = + −f x 10 1. x 1 3 Isolate the logarithm. Change to exponential form. Solve for y .
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