# 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.

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.

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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