SECTION 5.4 Logarithmic Functions 321 SUMMARY Properties of the Logarithmic Function ( ) = > f x x a log , 1 a ( ) = = y x x a log ifandonlyif a y • Domain: the interval ( )∞ 0, ; Range: the interval ( ) −∞ ∞, • x -intercept: 1; y -intercept: none; vertical asymptote: = x 0 ( y -axis); increasing; one-to-one • The graph of f is smooth and continuous. It contains the points ( ) ( ) − a 1 , 1 , 1,0 ,and ( ) a, 1 . • See Figure 44(a) for a typical graph. ( ) = < < f x x a log , 0 1 a ( ) = = y x x a log ifandonlyif a y • Domain: the interval ( )∞ 0, ; Range: the interval ( ) −∞ ∞, • x -intercept: 1; y -intercept: none; vertical asymptote: = x 0 ( y -axis); decreasing; one-to-one • The graph of f is smooth and continuous. It contains the points ( ) a, 1 , ( ) 1, 0 , and ( ) − a 1 , 1 . • See Figure 44(b) for a typical graph. Figure 44 x y 5 loga x y 23 3 (1, 0) (a, 1) , 21 1 a( (a, 1) 3 (b) 0 , a , 1 x y 23 3 (1, 0) 3 23 (a) a . 1 y 5 loga x , 21) 1 a( x 5 0 x 5 0 23 ) ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 5.4 Assess Your Understanding 1. Solve each inequality: (a) − ≤ − x x 3 7 8 2 (pp. A80–A81) (b) − − > x x 6 0 2 (pp. 180–182) 2. Solve the inequality: − + > x x 1 4 0 (pp. 261–263) Concepts and Vocabulary 3. Interactive Figure Exercise Exploring Logarithmic Functions Open the “Logarithmic Functions” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) The interactive figure shows, the graph of ( ) ( ) = ⋅ − + f x c x h k loga Set the values of c to 1, h to 0, and k to 0. Use the slider to adjust the value of a from 0.25 to 3. A logarithmic function is (increasing/decreasing) if < < a 0 1 and is (increasing/decreasing) if > a 1. (b) Set the values of a to 2, c to 1, h to 0, and k to 0.The graph of ( ) = f x x log2 contains the points ( ) ( ) 1 2 , , 1, , and ( ) 2, . (c) Set the values of a to 3, c to 1, h to 0, and k to 0.The graph of ( ) = f x x log3 contains the points ( ) ( ) 1 3 , , 1, , and ( ) 3, . 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure (continued)
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