1028 3 342 2 f 342 f 7 3 1 f g 7 f g 7 108 Therefore 7 3 1 344 g f 3 3 3 2 3 1

1028 3 342 2 f 342 f 7 3 1 f g 7 f g 7 108 therefore

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1028 3 342 2 f 342 f 7 3 1 f g 7 f g 7 108. Therefore, 7 3 1 344. g f 3 3 3 2 3 1 g f ( x g f x g 3 x 2 3 x 2 3 1 (b) Increasing on Decreasing on (c) Relative minimum: 0, 0 , 0 0,
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Section 3.3 Properties of Logarithms 281 12. log 1 4 5 log 5 log 1 4 ln 5 ln 1 4 1.161 13. log 9 0.4 log 0.4 log 9 ln 0.4 ln 9 0.417 14. log 20 0.125 log 0.125 log 20 ln 0.125 ln 20 0.694 15. log 15 1250 log 1250 log 15 ln 1250 ln 15 2.633 16. log 3 0.015 log 0.015 log 3 ln 0.015 ln 3 3.823 17. log 4 8 log 2 8 log 2 4 log 2 2 3 log 2 2 2 3 2 Section 3.3 Properties of Logarithms You should know the following properties of logarithms. (a) (b) (c) (d) You should be able to rewrite logarithmic expressions using these properties. ln u n n ln u log a u n n log a u ln u v ln u ln v log a u v log a u log a v ln uv ln u ln v log a uv log a u log a v log a x ln x ln a log a x log 10 x log 10 a log a x log b x log b a Vocabulary Check 1. change-of-base 2. 3. 4. This is the Product Property. Matches (c). This is the Power Property. Matches (a). 5. This is the Quotient Property. Matches (b). log a u v log a u log a v ln u n n ln u log a uv log a u log a v log x log a ln x ln a 1. (a) (b) log 5 x ln x ln 5 log 5 x log x log 5 2. (a) (b) log 3 x ln x ln 3 log 3 x log x log 3 3. (a) (b) log 1 5 x ln x ln 1 5 log 1 5 x log x log 1 5 4. (a) (b) log 1 3 x ln x ln 1 3 log 1 3 x log x log 1 3 5. (a) (b) log x 3 10 ln 3 10 ln x log x 3 10 log 3 10 log x 6. (a) (b) log x 3 4 ln 3 4 ln x log x 3 4 log 3 4 log x 7. (a) (b) log 2.6 x ln x ln 2.6 log 2.6 x log x log 2.6 8. (a) (b) log 7.1 x ln x ln 7.1 log 7.1 x log x log 7.1 9. log 3 7 log 7 log 3 ln 7 ln 3 1.771 10. log 7 4 log 4 log 7 ln 4 ln 7 0.712 11. log 1 2 4 log 4 log 1 2 ln 4 ln 1 2 2.000
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282 Chapter 3 Exponential and Logarithmic Functions 18. 4 4 log 2 3 4 log 2 2 4 log 2 3 2 log 2 2 2 4 log 2 3 2 log 2 4 4 log 2 3 log 2 4 2 3 4 log 2 4 2 log 2 3 4 19. 3 log 5 2 log 5 5 3 log 5 2 1 log 5 1 125 log 5 1 2 log 5 1 250 log 5 1 125 1 2 20. log 3 2 log 3 2 log 10 log 3 log 10 2 log 3 log 100 log 9 300 log 3 100 21. 6 ln 5 ln 5 6 ln 5 e 6 ln 5 ln e 6 22. ln 6 2 ln 6 2 ln e ln 6 e 2 ln 6 ln e 2 23. log 3 9 2 log 3 3 2 24. log 5 1 125 log 5 5 3 3 log 5 5 3 1 3 25. log 2 4 8 1 4 log 2 2 3 3 4 log 2 2 3 4 1 3 4 26. log 6 3 6 log 6 6 1 3 1 3 log 6 6 1 3 1 1 3 27. log 4 16 1.2 1.2 log 4 16 1.2 log 4 4 2 1.2 2 2.4 28. 0.2 4 0.8 0.2 log 3 3 4 log 3 81 0.2 0.2 log 3 81 29. is undefined. is not in the domain of log 3 x . 9 log 3 9 30. is undefined because is not in the domain of log 2 x . 16 log 2 16 31. ln e 4.5 4.5 32. 12 12 1 3 ln e 4 3 4 ln e 33. 1 2 0 1 2 1 0 1 2 ln e ln 1 e ln 1 ln e 34. 3 4 3 4 1 3 4 ln e ln 4 e 3 ln e 3 4 35. ln e 2 ln e 5 2 5 7 36. 7 ln e 7 ln e 12 e 5 2 ln e 6 ln e 5 ln e 12 ln e 5 37. 2 2 log 5 5 log 5 5 2 log 5 25 log 5 75 log 5 3 log 5 75 3 38. 3 1 2 1 5 2 1 1 2 log 4 4 5 2 log 4 4 log 4 2 log 4 32 log 4 4 1 2 log 4 4 5 2 39. log 4 5 x log 4 5 log 4 x
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Section 3.3 Properties of Logarithms 283 55. 4 ln x 1 2 ln y 5 ln z ln x 4 ln y ln z 5 ln x 4 y z 5 ln x 4 y ln z 5 56.
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