1. How does the power rule for logarithms help when solving logarithms with the form [latex]{\mathrm{log}}_{b}\left(\sqrt[n]{x}\right)\\[/latex]?

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

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

3. [latex]{\mathrm{log}}_{b}\left(7x\cdot 2y\right)\\[/latex]

4. [latex]\mathrm{ln}\left(3ab\cdot 5c\right)\\[/latex]

5. [latex]{\mathrm{log}}_{b}\left(\frac{13}{17}\right)\\[/latex]

6. [latex]{\mathrm{log}}_{4}\left(\frac{\text{ }\frac{x}{z}\text{ }}{w}\right)\\[/latex]

7. [latex]\mathrm{ln}\left(\frac{1}{{4}^{k}}\right)\\[/latex]

8. [latex]{\mathrm{log}}_{2}\left({y}^{x}\right)\\[/latex]

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

9. [latex]\mathrm{ln}\left(7\right)+\mathrm{ln}\left(x\right)+\mathrm{ln}\left(y\right)\\[/latex]

10. [latex]{\mathrm{log}}_{3}\left(2\right)+{\mathrm{log}}_{3}\left(a\right)+{\mathrm{log}}_{3}\left(11\right)+{\mathrm{log}}_{3}\left(b\right)\\[/latex]

11. [latex]{\mathrm{log}}_{b}\left(28\right)-{\mathrm{log}}_{b}\left(7\right)\\[/latex]

12. [latex]\mathrm{ln}\left(a\right)-\mathrm{ln}\left(d\right)-\mathrm{ln}\left(c\right)\\[/latex]

13. [latex]-{\mathrm{log}}_{b}\left(\frac{1}{7}\right)\\[/latex]

14. [latex]\frac{1}{3}\mathrm{ln}\left(8\right)\\[/latex]

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. [latex]\mathrm{log}\left(\frac{{x}^{15}{y}^{13}}{{z}^{19}}\right)\\[/latex]

16. [latex]\mathrm{ln}\left(\frac{{a}^{-2}}{{b}^{-4}{c}^{5}}\right)\\[/latex]

17. [latex]\mathrm{log}\left(\sqrt{{x}^{3}{y}^{-4}}\right)\\[/latex]

18. [latex]\mathrm{ln}\left(y\sqrt{\frac{y}{1-y}}\right)\\[/latex]

19. [latex]\mathrm{log}\left({x}^{2}{y}^{3}\sqrt[3]{{x}^{2}{y}^{5}}\right)\\[/latex]

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

20. [latex]\mathrm{log}\left(2{x}^{4}\right)+\mathrm{log}\left(3{x}^{5}\right)\\[/latex]

21. [latex]\mathrm{ln}\left(6{x}^{9}\right)-\mathrm{ln}\left(3{x}^{2}\right)\\[/latex]

22. [latex]2\mathrm{log}\left(x\right)+3\mathrm{log}\left(x+1\right)\\[/latex]

23. [latex]\mathrm{log}\left(x\right)-\frac{1}{2}\mathrm{log}\left(y\right)+3\mathrm{log}\left(z\right)\\[/latex]

24. [latex]4{\mathrm{log}}_{7}\left(c\right)+\frac{{\mathrm{log}}_{7}\left(a\right)}{3}+\frac{{\mathrm{log}}_{7}\left(b\right)}{3}\\[/latex]

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

25. [latex]{\mathrm{log}}_{7}\left(15\right)\\[/latex] to base e

26. [latex]{\mathrm{log}}_{14}\left(55.875\right)\\[/latex] to base 10

For the following exercises, suppose [latex]{\mathrm{log}}_{5}\left(6\right)=a\\[/latex] and [latex]{\mathrm{log}}_{5}\left(11\right)=b\\[/latex]. 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. [latex]{\mathrm{log}}_{11}\left(5\right)\\[/latex]

28. [latex]{\mathrm{log}}_{6}\left(55\right)\\[/latex]

29. [latex]{\mathrm{log}}_{11}\left(\frac{6}{11}\right)\\[/latex]

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

30. [latex]{\mathrm{log}}_{3}\left(\frac{1}{9}\right)-3{\mathrm{log}}_{3}\left(3\right)\\[/latex]

31. [latex]6{\mathrm{log}}_{8}\left(2\right)+\frac{{\mathrm{log}}_{8}\left(64\right)}{3{\mathrm{log}}_{8}\left(4\right)}\\[/latex]

32. [latex]2{\mathrm{log}}_{9}\left(3\right)-4{\mathrm{log}}_{9}\left(3\right)+{\mathrm{log}}_{9}\left(\frac{1}{729}\right)\\[/latex]

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. [latex]{\mathrm{log}}_{3}\left(22\right)\\[/latex]

34. [latex]{\mathrm{log}}_{8}\left(65\right)\\[/latex]

35. [latex]{\mathrm{log}}_{6}\left(5.38\right)\\[/latex]

36. [latex]{\mathrm{log}}_{4}\left(\frac{15}{2}\right)\\[/latex]

37. [latex]{\mathrm{log}}_{\frac{1}{2}}\left(4.7\right)\\[/latex]

38. Use the product rule for logarithms to find all x values such that [latex]{\mathrm{log}}_{12}\left(2x+6\right)+{\mathrm{log}}_{12}\left(x+2\right)=2\\[/latex]. Show the steps for solving.

39. Use the quotient rule for logarithms to find all x values such that [latex]{\mathrm{log}}_{6}\left(x+2\right)-{\mathrm{log}}_{6}\left(x - 3\right)=1\\[/latex]. Show the steps for solving.

40. Can the power property of logarithms be derived from the power property of exponents using the equation [latex]{b}^{x}=m?\\[/latex] If not, explain why. If so, show the derivation.

41. Prove that [latex]{\mathrm{log}}_{b}\left(n\right)=\frac{1}{{\mathrm{log}}_{n}\left(b\right)}\\[/latex] for any positive integers b > 1 and n > 1.

42. Does [latex]{\mathrm{log}}_{81}\left(2401\right)={\mathrm{log}}_{3}\left(7\right)\\[/latex]? Verify the claim algebraically.