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Study Guides > College Algebra

Section Exercises

1. How does the power rule for logarithms help when solving logarithms with the form logb(xn){\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. logb(7x2y){\mathrm{log}}_{b}\left(7x\cdot 2y\right) 4. ln(3ab5c)\mathrm{ln}\left(3ab\cdot 5c\right) 5. logb(1317){\mathrm{log}}_{b}\left(\frac{13}{17}\right) 6. log4( xz w){\mathrm{log}}_{4}\left(\frac{\text{ }\frac{x}{z}\text{ }}{w}\right) 7. ln(14k)\mathrm{ln}\left(\frac{1}{{4}^{k}}\right) 8. log2(yx){\mathrm{log}}_{2}\left({y}^{x}\right) For the following exercises, condense to a single logarithm if possible. 9. ln(7)+ln(x)+ln(y)\mathrm{ln}\left(7\right)+\mathrm{ln}\left(x\right)+\mathrm{ln}\left(y\right) 10. log3(2)+log3(a)+log3(11)+log3(b){\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. logb(28)logb(7){\mathrm{log}}_{b}\left(28\right)-{\mathrm{log}}_{b}\left(7\right) 12. ln(a)ln(d)ln(c)\mathrm{ln}\left(a\right)-\mathrm{ln}\left(d\right)-\mathrm{ln}\left(c\right) 13. logb(17)-{\mathrm{log}}_{b}\left(\frac{1}{7}\right) 14. 13ln(8)\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. log(x15y13z19)\mathrm{log}\left(\frac{{x}^{15}{y}^{13}}{{z}^{19}}\right) 16. ln(a2b4c5)\mathrm{ln}\left(\frac{{a}^{-2}}{{b}^{-4}{c}^{5}}\right) 17. log(x3y4)\mathrm{log}\left(\sqrt{{x}^{3}{y}^{-4}}\right) 18. ln(yy1y)\mathrm{ln}\left(y\sqrt{\frac{y}{1-y}}\right) 19. log(x2y3x2y53)\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. log(2x4)+log(3x5)\mathrm{log}\left(2{x}^{4}\right)+\mathrm{log}\left(3{x}^{5}\right) 21. ln(6x9)ln(3x2)\mathrm{ln}\left(6{x}^{9}\right)-\mathrm{ln}\left(3{x}^{2}\right) 22. 2log(x)+3log(x+1)2\mathrm{log}\left(x\right)+3\mathrm{log}\left(x+1\right) 23. log(x)12log(y)+3log(z)\mathrm{log}\left(x\right)-\frac{1}{2}\mathrm{log}\left(y\right)+3\mathrm{log}\left(z\right) 24. 4log7(c)+log7(a)3+log7(b)34{\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. log7(15){\mathrm{log}}_{7}\left(15\right) to base e 26. log14(55.875){\mathrm{log}}_{14}\left(55.875\right) to base 10 For the following exercises, suppose log5(6)=a{\mathrm{log}}_{5}\left(6\right)=a and log5(11)=b{\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. log11(5){\mathrm{log}}_{11}\left(5\right) 28. log6(55){\mathrm{log}}_{6}\left(55\right) 29. log11(611){\mathrm{log}}_{11}\left(\frac{6}{11}\right) For the following exercises, use properties of logarithms to evaluate without using a calculator. 30. log3(19)3log3(3){\mathrm{log}}_{3}\left(\frac{1}{9}\right)-3{\mathrm{log}}_{3}\left(3\right) 31. 6log8(2)+log8(64)3log8(4)6{\mathrm{log}}_{8}\left(2\right)+\frac{{\mathrm{log}}_{8}\left(64\right)}{3{\mathrm{log}}_{8}\left(4\right)} 32. 2log9(3)4log9(3)+log9(1729)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. log3(22){\mathrm{log}}_{3}\left(22\right) 34. log8(65){\mathrm{log}}_{8}\left(65\right) 35. log6(5.38){\mathrm{log}}_{6}\left(5.38\right) 36. log4(152){\mathrm{log}}_{4}\left(\frac{15}{2}\right) 37. log12(4.7){\mathrm{log}}_{\frac{1}{2}}\left(4.7\right) 38. Use the product rule for logarithms to find all x values such that log12(2x+6)+log12(x+2)=2{\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 log6(x+2)log6(x3)=1{\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 bx=m?{b}^{x}=m? If not, explain why. If so, show the derivation. 41. Prove that logb(n)=1logn(b){\mathrm{log}}_{b}\left(n\right)=\frac{1}{{\mathrm{log}}_{n}\left(b\right)} for any positive integers > 1 and > 1. 42. Does log81(2401)=log3(7){\mathrm{log}}_{81}\left(2401\right)={\mathrm{log}}_{3}\left(7\right)? Verify the claim algebraically.

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