What does the change-of-base formula do? Why is it useful when using a calculator? AlgebRAIC For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

sectioN exercises 515

49. Use f (x) = log4(x) as the parent function.

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1 2 3 4 5

321 4 5

50. Use f (x) = log5(x) as the parent function.

x

y

–1–2–3–4–5 –1 –2 –3 –4 –5

1 2 3 4 5

321 4 5

TeChnOlOgy

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

51. log(x − 1) + 2 = ln(x − 1) + 2 52. log(2x − 3) + 2 = −log(2x − 3) + 5 53. ln(x − 2) = −ln(x + 1)

54. 2ln(5x + 1) = 1 _ 2 ln(−5x) + 1 55. 1 _ 3 log(1 − x) = log(x + 1) +

1 _ 3

exTenSIOnS 56. Let b be any positive real number such that b ≠ 1.

What must logb1 be equal to? Verify the result. 57. Explore and discuss the graphs of f (x) = log

1 _ 2 (x)

and g(x) = −log2(x). Make a conjecture based on the result.

58. Prove the conjecture made in the previous exercise. 59. What is the domain of the function

f (x) = ln  x + 2 _ x − 4  ? Discuss the result.

60. Use properties of exponents to find the x-intercepts of the function f (x) = log(x 2 + 4x + 4) algebraically. Show the steps for solving, and then verify the result by graphing the function.

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SECTION 6.5 sectioN exercises 525

6.5 SeCTIOn exeRCISeS

veRbAl 1. How does the power rule for logarithms help when

solving logarithms with the form logb( n √

— x )?

2. What does the change-of-base formula do? Why is it useful when using a calculator?

AlgebRAIC For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

3. logb(7x · 2y) 4. ln(3ab · 5c) 5. logb  13 _ 17  6. log4 

x __ z _ w  7. ln  1 _ 4k  8. log2(yx) For the following exercises, condense to a single logarithm if possible.

9. ln(7) + ln(x) + ln(y) 10. log3(2) + log3(a) + log3(11) + log3(b) 11. logb(28) − logb(7)

12. ln(a) − ln(d) − ln(c) 13. −logb  1 _ 7  14. 1 _ 3 ln(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.

15. log  x15 y13 _ z19  16. ln  a −2 _

b−4 c5  17. log( √— x3 y−4 ) 18. ln  y √

_____

y _ 1 − y  19. log(x 2 y 3

3 √ —

x2 y5 )

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. 20. log(2×4) + log(3×5) 21. ln(6×9) − ln(3×2) 22. 2log(x) + 3log(x + 1)

23. log(x) − 1 _ 2 log(y) + 3log(z) 24. 4log7 (c) +

log7(a) _ 3 + log7(b) _ 3

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base. 25. log7(15) to base e 26. log14(55.875) to base 10

For the following exercises, suppose log5 (6) = a and log5 (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. log11(5) 28. log6(55) 29. log11  6 _ 11  nUmeRIC For the following exercises, use properties of logarithms to evaluate without using a calculator.

30. log3  1 _ 9  − 3log3 (3) 31. 6log8(2) + log8(64) _ 3log8(4)

32. 2log9(3) − 4log9(3) + log9  1 _ 729  For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

33. log3(22) 34. log8(65) 35. log6(5.38) 36. log4  15 _ 2  37. log 1 _ 2 (4.7)

exTenSIOnS 38. Use the product rule for logarithms to find all x

values such that log12(2x + 6) + log12(x + 2) = 2. Show the steps for solving.

39. Use the quotient rule for logarithms to find all x values such that log6(x + 2) − log6 (x − 3) = 1. Show the steps for solving.

40. Can the power property of logarithms be derived from the power property of exponents using the equation b x = m? If not, explain why. If so, show the derivation.

41. Prove that logb (n) = 1 _

logn(b) for any positive integers

b > 1 and n > 1.

42. Does log81(2401) = log3(7)? Verify the claim algebraically.

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