SECTION 5.4 Logarithmic Functions 315 The domain of a logarithmic function consists of the positive real numbers, so the argument of a logarithmic function must be greater than zero. Finding the Domain of a Logarithmic Function Find the domain of each logarithmic function. (a) ( ) ( ) = + F x x log 3 2 (b) ( ) ( ) = + − g x x x log 1 1 5 (c) ( ) = h x x log 1 2 Solution EXAMPLE 5 (a) The domain of F consists of all x for which + > x 3 0, that is, >− x 3. Using interval notation, the domain of F is ( ) − ∞ 3, . (b) The domain of g is restricted to + − > x x 1 1 0 Solve this inequality to find that the domain of g consists of all x between −1 and 1, that is, − < < x 1 1, or, using interval notation, ( ) −1, 1 . (c) Since > x 0, provided that ≠ x 0, the domain of h consists of all real numbers except zero, or, using interval notation, ( ) ( ) −∞ ∪ ∞ , 0 0, . Now Work PROBLEMS 43 AND 49 4 Graph Logarithmic Functions Because exponential functions and logarithmic functions are inverses of each other, the graph of the logarithmic function = y x loga is the reflection about the line = y x of the graph of the exponential function = y a ,x as shown in Figures 35(a) and (b). Figure 35 x y 5 loga x y 5 x y 5 a x y 23 3 (1, 0) (a, 1) (a, 1) (0, 1) (1, a) (1, a) 3 23 (a) 0 , a , 1 x y 5 x y 23 3 (1, 0) (0, 1) 3 23 (b) a . 1 y 5 loga x y 5 a x ( 21, ) 1 – a ( ,21) 1 – a ( , 21) 1 – a (21, ) 1 – a For example, to graph = y x log , 2 graph = y 2x and reflect it about the line = y x. See Figure 36.To graph = y x log , 1 3 graph ( ) = y 1 3 x and reflect it about the line = y x. See Figure 37. Figure 36 (0, 1) x y 22 2 2 22 (1, 0) (1, 2) (2, 1) ( ) 1 2 , 21 ( ) 1 2 21, y 5 x y 5 2 x y 5 log2x Figure 37 (21, 3) (3, 21) x 23 3 23 (1, 0) (0, 1) ( ) 1 3 ( ) 1 3 1, ( ) y 5 x y 5 y 5 log1/3x x 3 1 3 , 1 y Now Work PROBLEM 63
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