322 CHAPTER 5 Exponential and Logarithmic Functions (e) Set the values of a to 0.5, c to 1, h to 0, and k to 0. Check the “Show Inverse Function” box. Use the slider to adjust the value of h from −3 to 3 and pay attention to the vertical asymptote of the logarithmic function and the horizontal asymptote of the exponential function. If ( ) ( ) = + − f x x 2 log 2 1, 3 what would the horizontal asymptote of the inverse exponential function be? (f) If < c 0 and < < a 0 1, the logarithmic function ( ) ( ) = ⋅ − + f x c x h k loga is (increasing/ decreasing) over its domain. (g) Set the values of a to 2, c to 1, h to 0 and k to 0. Use the slider to adjust the value of h from −3 to 3 and pay attention to the equation of the logarithmic function. Note the domain for each value of h. Based on what you observe, what would be the domain of ( ) ( ) = − f x x 3log 5 . 2 What would be the range of −f ?1 Express your answers in interval notation. 5. The domain of the logarithmic function ( ) = f x x loga is . 6. The graph of every logarithmic function ( ) = f x x log , a where > a 0 and ≠ a 1, contains the three points: ( ) −, 1, ( ) , 0 , and ( ) , 1 . 7. If the graph of a logarithmic function ( ) = f x x log , a where > a 0 and ≠ a 1, is increasing, then its base is larger than . 8. True or False If = y x log , a then = y a . x 9. True or False The graph of ( ) = f x x log , a where > a 0 and ≠ a 1, has an x-intercept equal to 1 and no y-intercept. 10. Multiple Choice Select the answer that completes the statement: = y x ln if and only if . (a) = x ey (b) = y ex (c) = x 10y (d) = y 10x 11. Multiple Choice The domain of ( ) ( ) = + f x x log 2 3 is (a) ( ) −∞ ∞, (b) ( )∞ 2, (c) ( ) − ∞ 2, (d) ( )∞ 0, 12. Multiple Choice log 81 3 equals (a) 9 (b) 4 (c) 2 (d) 3 (d) Set the values of a to ( ) = 0.5 1 2 , c to 1, h to 0, and k to 0. The graph of ( ) = f x x log1 2 contains the points ( ) ( ) 1 2 , , 1, , and ( ) 2, . 4. 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) Set the values of a to 2, c to 1, h to 0, and k to 0. Use the slider to adjust the value of h from −3 to 3. Pay attention to the equation of the function. Based on your observations, the graph of ( ) ( ) = − g x x log 5 2 is obtained from the graph of ( ) = f x x log2 by shifting the graph of ( ) = f x x log2 (horizontally/ vertically), five units (right/left/up/down). (b) Set the values of a to 2, c to 1, h to 0, and k to 0. Use the slider to adjust the value of h from −3 to 3. Pay attention to the equation of the function. Based on your observations, the graph of ( ) ( ) = + g x x log 7 2 is obtained from the graph of ( ) = f x x log2 by shifting the graph of ( ) = f x x log2 (horizontally/ vertically), seven units (right/left/up/down). (c) Set the values of a to 2, c to 1, h to 0, and k to 0. Use the slider to adjust the value of h from −3 to 3. Pay attention to the equation of the function and the vertical asymptote. Based on your observations, what would the vertical asymptote of ( ) ( ) = − f x x 2 log 5 2 be? (d) Set the values of a to 2, c to 1, h to 0, and k to 0. Use the slider to adjust the value of k from −3 to 3. Pay attention to the equation of the function. Based on your observations, the graph of ( ) = + g x x log 5 2 is obtained from the graph of ( ) = f x x log2 by shifting the graph of ( ) = f x x log2 (horizontally/vertically), five units (right/left/up/down). Skill Building In Problems 13–20, change each exponential statement to an equivalent statement involving a logarithm. 13. = 9 32 14. = 16 42 15. = a 1.6 2 16. = a 2.1 3 17. = 2 7.2 x 18. = 3 4.6 x 19. = e 8 x 20. = e M 2.2 In Problems 21–28, change each logarithmic statement to an equivalent statement involving an exponent. 21. = log 8 3 2 22. ( ) = − log 1 9 2 3 23. = log 3 6 a 24. = log 4 2 b 25. = x log 23 26. = x log 62 27. = x ln4 28. = x ln 4 In Problems 29–40, find the exact value of each logarithm without using a calculator. 29. log 12 30. log 88 31. log 49 7 32. ( ) log 1 9 3 33. log 125 1 5 34. log 9 1 3 35. log 10 36. log 100 3 37. log 42 38. log 93 39. e ln 40. e ln 3
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