Section Exercises

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

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. ${\mathrm{log}}_{b}\left(7x\cdot 2y\right)$

4. $\mathrm{ln}\left(3ab\cdot 5c\right)$

5. ${\mathrm{log}}_{b}\left(\frac{13}{17}\right)$

6. ${\mathrm{log}}_{4}\left(\frac{\text{ }\frac{x}{z}\text{ }}{w}\right)$

7. $\mathrm{ln}\left(\frac{1}{{4}^{k}}\right)$

8. ${\mathrm{log}}_{2}\left({y}^{x}\right)$

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

9. $\mathrm{ln}\left(7\right)+\mathrm{ln}\left(x\right)+\mathrm{ln}\left(y\right)$

10. ${\mathrm{log}}_{3}\left(2\right)+{\mathrm{log}}_{3}\left(a\right)+{\mathrm{log}}_{3}\left(11\right)+{\mathrm{log}}_{3}\left(b\right)$

11. ${\mathrm{log}}_{b}\left(28\right)-{\mathrm{log}}_{b}\left(7\right)$

12. $\mathrm{ln}\left(a\right)-\mathrm{ln}\left(d\right)-\mathrm{ln}\left(c\right)$

13. $-{\mathrm{log}}_{b}\left(\frac{1}{7}\right)$

14. $\frac{1}{3}\mathrm{ln}\left(8\right)$

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. $\mathrm{log}\left(\frac{{x}^{15}{y}^{13}}{{z}^{19}}\right)$

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

17. $\mathrm{log}\left(\sqrt{{x}^{3}{y}^{-4}}\right)$

18. $\mathrm{ln}\left(y\sqrt{\frac{y}{1-y}}\right)$

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

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

20. $\mathrm{log}\left(2{x}^{4}\right)+\mathrm{log}\left(3{x}^{5}\right)$

21. $\mathrm{ln}\left(6{x}^{9}\right)-\mathrm{ln}\left(3{x}^{2}\right)$

22. $2\mathrm{log}\left(x\right)+3\mathrm{log}\left(x+1\right)$

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

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

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

25. ${\mathrm{log}}_{7}\left(15\right)$ to base e

26. ${\mathrm{log}}_{14}\left(55.875\right)$ to base 10

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

28. ${\mathrm{log}}_{6}\left(55\right)$

29. ${\mathrm{log}}_{11}\left(\frac{6}{11}\right)$

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

30. ${\mathrm{log}}_{3}\left(\frac{1}{9}\right)-3{\mathrm{log}}_{3}\left(3\right)$

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

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

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.

33. ${\mathrm{log}}_{3}\left(22\right)$

34. ${\mathrm{log}}_{8}\left(65\right)$

35. ${\mathrm{log}}_{6}\left(5.38\right)$

36. ${\mathrm{log}}_{4}\left(\frac{15}{2}\right)$

37. ${\mathrm{log}}_{\frac{1}{2}}\left(4.7\right)$

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

39. Use the quotient rule for logarithms to find all x values such that ${\mathrm{log}}_{6}\left(x+2\right)-{\mathrm{log}}_{6}\left(x - 3\right)=1$. 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 ${\mathrm{log}}_{b}\left(n\right)=\frac{1}{{\mathrm{log}}_{n}\left(b\right)}$ for any positive integers > 1 and > 1.

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