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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.