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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