SECTION 4.5 SECTION EXERCISES 389 4.5 SECTION EXERCISES VERBAL as the quotient of 1. How does the power rule for logarithms help when solving logarithms with the form log, (Wx)? 2. What does the change-of-base formula do? Why is it useful when using a calculator? ALGEBRAIC or product of logs. For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, 3. log, (7x . 2y) ient of logs with any 4. In(3ab . 5c) 5.1986 (17 ) 6. log. Z atural log, In(x), has TAUDS DIMHTIRA 8. logz (y' ) For the following exercises, condense to a single logarithm if possible. 9. In( 7) + In(x) + In(y) 10. log, (2) + log, (a) + log,(11) + log,(b) 11. log,(28) - log,(7) 12. In(a) - In(d) - In(c) 13. Togi -) 14 . - In(8 ) For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. n = e. 15. log 16. In a-2 16 - 4 C 5 17. log ( V x y- 4 ) 18. In ( WV ILY) 19 . log(x - y3 Vx ys ) will be the natural For the following exercises, condense each expression to a single logarithm using the properties of logarithms. 20. log(2x*) + log(3x5) 21. In (6x') - In(3x2) 22. 2log(x) + 3log(x + 1) log,(a) log, (b) 23. log(x) - - log(y) + 3log(z) 24. 4log, (C) + 2 For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base. 25. log, (15 ) to base e 26. log (55.875) to base 10 For the following exercises, suppose log, (6) = a and log, (11) = b. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a and b. Show the steps for solving. 27. 10g1, (5) 28. log (55) 29. 10841 11) NUMERIC For the following exercises, use properties of logarithms to evaluate without using a calculator. logg (64) 30. log, 9 - 31083 (3 ) 31. 610g (2) + 310gg (4) 32. 2log,(3) - 410g,(3) + log, 729) For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. f any other base. Use a calculator to approximate each to five decimal places. is the log base e. 33. log, (22) 34. 1ogg (65) 35. log (5.38) 36. log, 2 37, log, ( 4.7 ) EXTENSIONS 38. Use the product rule for logarithms to find all x 39. Use the quotient rule for logarithms to find all x values such that log,2(2x + 6) + log,2 (x + 2) = 2. values such that log (x + 2) - log. (x - 3) = 1. Show Show the steps for solving. the steps for solving. 40. Can the power property of logarithms be derived 41. Prove that log, (n) = 1 for any positive integers log (b) from the power property of exponents using the 6 > 1 and n > 1. equation b* = m? If not, explain why. If so, show the derivation. 42. Does logs (2401) = log,(7)? Verify the claim algebraically