13.5.2: Homework

Algebraic

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

14. \(\log_b ( 7x \cdot 2y )\)
15. \(\ln( 3ab \cdot 5c )\)
16. \(\log_b \left( \frac{13}{17} \right)\)
17. \( \log_4 \left( \frac{x/z}{w} \right)\)
18. \(\ln\left( \frac{1}{4^k} \right)\)
19. \( \log_2 ( y^x )\)

For the following exercises, condense to a single logarithm if possible.

20. \(\ln( 7 )+\ln( x )+\ln( y )\)
21. \(\log_3 (2)+\log_3 (a)+\log_3 (11)+\log_3 (b)\)
22. \(\log_b (28)−\log_b (7)\)
23. \(\ln( a )−\ln( d )−\ln( c )\)
24. \(−\log_b \left( \frac{1}{7} \right)\)
25. \(\frac{1}{3} \ln( 8 )\)

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.

26. \(\log\left( \frac{x^{15} y^{13}}{z^{19}} \right)\)
27. \(\ln\left( \frac{a^{-2}}{b^{-4} c^5} \right)\)
28. \(\log( \sqrt{ x^3 y^{−4} } )\)
29. \(\ln\left( y \sqrt{\frac{y}{1 - y}} \right)\)
30. \(\log( x^2 y^3 \sqrt[3]{x^2 y^5} )\)

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.

31. \(\log( 2 x^4 )+\log( 3 x^5 )\)
32. \(\ln(6 x^9 )−\ln(3 x^2 )\)
33. \(2\log(x)+3\log(x+1)\)
34. \(\log(x)− \frac{1}{2}\log(y)+3\log(z)\)
35. \(4\log_7 ( c ) + \frac{\log_7 ( a )}{3} + \frac{\log_7 ( b )}{3}\)

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.

36. \(\log_7 ( 15 )\) to base \(e\)
37. \(\log_{14} ( 55.875 )\) to base \(10\)

For the following exercises, suppose \(\log_5 ( 6 )=a\) and \(\log_5 ( 11 )=b\). 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.

38. \(\log_{11} ( 5 )\)
39. \(\log_6 ( 55 )\)
40. \(\log_{11} \left( \frac{6}{11} \right)\)

Numeric

For the following exercises, use properties of logarithms to evaluate without using a calculator.

41. \(\log_3 \left( \frac{1}{9} \right)−3\log_3 ( 3 )\)
42. \(6\log_8 ( 2 )+ \frac{\log_8 ( 64 )}{3\log_8 ( 4 )}\)
43. \(2\log_9 ( 3 )−4\log_9 ( 3 )+\log_9 \left( \frac{1}{729} \right)\)

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.

44. \(\log_3 ( 22 )\)
45. \(\log_8 ( 65 )\)
46. \(\log_6 ( 5.38 )\)
47. \(\log_4 \left( \frac{15}{2} \right)\)
48. \(\log_{1/2} ( 4.7 )\)

Extensions

49. Use the Product Law of Logarithms to find all \(x\) values such that \(\log_{12} ( 2x+6 )+\log_{12} ( x+2 )=2\). Show the steps for solving.
50. Use the Quotient Law of Logarithms to find all \(x\) values such that \(\log_6 ( x+2 )−\log_6 ( x−3 )=1\). Show the steps for solving.
51. Can the Power Law of Logarithms be derived from the Power Law of Exponents using the equation \(b^x =m\)? If not, explain why. If so, show the derivation.
52. Prove that \(\log_b ( n )= \frac{1}{\log_n ( b )}\) for any positive integers \(b>1\) and \(n>1\).
53. Does \(\log_{81} ( 2401 )=\log_3 ( 7 )\)? Verify the claim algebraically.