{"id":11216,"date":"2015-07-14T18:52:56","date_gmt":"2015-07-14T18:52:56","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11216"},"modified":"2021-11-15T01:43:15","modified_gmt":"2021-11-15T01:43:15","slug":"problem-set-logarithmic-properties","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/problem-set-logarithmic-properties\/","title":{"raw":"Problem Set: Logarithmic Properties","rendered":"Problem Set: Logarithmic Properties"},"content":{"raw":"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]?\r\n\r\n2.\u00a0What does the change-of-base formula do? Why is it useful when using a calculator?\r\n\r\nFor the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.\r\n\r\n3.\u00a0[latex]{\\mathrm{log}}_{b}\\left(7x\\cdot 2y\\right)[\/latex]\r\n\r\n4.\u00a0[latex]\\mathrm{ln}\\left(3ab\\cdot 5c\\right)[\/latex]\r\n\r\n5. [latex]{\\mathrm{log}}_{b}\\left(\\frac{13}{17}\\right)[\/latex]\r\n\r\n6.\u00a0[latex]{\\mathrm{log}}_{4}\\left(\\frac{\\text{ }\\frac{x}{z}\\text{ }}{w}\\right)[\/latex]\r\n\r\n7. [latex]\\mathrm{ln}\\left(\\frac{1}{{4}^{k}}\\right)[\/latex]\r\n\r\n8.\u00a0[latex]{\\mathrm{log}}_{2}\\left({y}^{x}\\right)[\/latex]\r\n\r\nFor the following exercises, condense to a single logarithm if possible.\r\n\r\n9. [latex]\\mathrm{ln}\\left(7\\right)+\\mathrm{ln}\\left(x\\right)+\\mathrm{ln}\\left(y\\right)[\/latex]\r\n\r\n10.\u00a0[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]\r\n\r\n11. [latex]{\\mathrm{log}}_{b}\\left(28\\right)-{\\mathrm{log}}_{b}\\left(7\\right)[\/latex]\r\n\r\n12.\u00a0[latex]\\mathrm{ln}\\left(a\\right)-\\mathrm{ln}\\left(d\\right)-\\mathrm{ln}\\left(c\\right)[\/latex]\r\n\r\n13. [latex]-{\\mathrm{log}}_{b}\\left(\\frac{1}{7}\\right)[\/latex]\r\n\r\n14.\u00a0[latex]\\frac{1}{3}\\mathrm{ln}\\left(8\\right)[\/latex]\r\n\r\nFor 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.\r\n\r\n15. [latex]\\mathrm{log}\\left(\\frac{{x}^{15}{y}^{13}}{{z}^{19}}\\right)[\/latex]\r\n\r\n16.\u00a0[latex]\\mathrm{ln}\\left(\\frac{{a}^{-2}}{{b}^{-4}{c}^{5}}\\right)[\/latex]\r\n\r\n17. [latex]\\mathrm{log}\\left(\\sqrt{{x}^{3}{y}^{-4}}\\right)[\/latex]\r\n\r\n18.\u00a0[latex]\\mathrm{ln}\\left(y\\sqrt{\\frac{y}{1-y}}\\right)[\/latex]\r\n\r\n19. [latex]\\mathrm{log}\\left({x}^{2}{y}^{3}\\sqrt[3]{{x}^{2}{y}^{5}}\\right)[\/latex]\r\n\r\nFor the following exercises, condense each expression to a single logarithm using the properties of logarithms.\r\n\r\n20. [latex]\\mathrm{log}\\left(2{x}^{4}\\right)+\\mathrm{log}\\left(3{x}^{5}\\right)[\/latex]\r\n\r\n21. [latex]\\mathrm{ln}\\left(6{x}^{9}\\right)-\\mathrm{ln}\\left(3{x}^{2}\\right)[\/latex]\r\n\r\n22.\u00a0[latex]2\\mathrm{log}\\left(x\\right)+3\\mathrm{log}\\left(x+1\\right)[\/latex]\r\n\r\n23. [latex]\\mathrm{log}\\left(x\\right)-\\frac{1}{2}\\mathrm{log}\\left(y\\right)+3\\mathrm{log}\\left(z\\right)[\/latex]\r\n\r\n24.\u00a0[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]\r\n\r\nFor the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.\r\n\r\n25. [latex]{\\mathrm{log}}_{7}\\left(15\\right)[\/latex] to base\u00a0e<\/em>\r\n\r\n26.\u00a0[latex]{\\mathrm{log}}_{14}\\left(55.875\\right)[\/latex] to base 10\r\n\r\nFor 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<\/em>\u00a0and b<\/em>. Show the steps for solving.\r\n\r\n27. [latex]{\\mathrm{log}}_{11}\\left(5\\right)[\/latex]\r\n\r\n28.\u00a0[latex]{\\mathrm{log}}_{6}\\left(55\\right)[\/latex]\r\n\r\n29. [latex]{\\mathrm{log}}_{11}\\left(\\frac{6}{11}\\right)[\/latex]\r\n\r\nFor the following exercises, use properties of logarithms to evaluate without using a calculator.\r\n\r\n30. [latex]{\\mathrm{log}}_{3}\\left(\\frac{1}{9}\\right)-3{\\mathrm{log}}_{3}\\left(3\\right)[\/latex]\r\n\r\n31. [latex]6{\\mathrm{log}}_{8}\\left(2\\right)+\\frac{{\\mathrm{log}}_{8}\\left(64\\right)}{3{\\mathrm{log}}_{8}\\left(4\\right)}[\/latex]\r\n\r\n32.\u00a0[latex]2{\\mathrm{log}}_{9}\\left(3\\right)-4{\\mathrm{log}}_{9}\\left(3\\right)+{\\mathrm{log}}_{9}\\left(\\frac{1}{729}\\right)[\/latex]\r\n\r\nFor 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.\r\n\r\n33. [latex]{\\mathrm{log}}_{3}\\left(22\\right)[\/latex]\r\n\r\n34.\u00a0[latex]{\\mathrm{log}}_{8}\\left(65\\right)[\/latex]\r\n\r\n35. [latex]{\\mathrm{log}}_{6}\\left(5.38\\right)[\/latex]\r\n\r\n36.\u00a0[latex]{\\mathrm{log}}_{4}\\left(\\frac{15}{2}\\right)[\/latex]\r\n\r\n37. [latex]{\\mathrm{log}}_{\\frac{1}{2}}\\left(4.7\\right)[\/latex]\r\n\r\n38. Use the product rule for logarithms to find all x<\/em>\u00a0values such that [latex]{\\mathrm{log}}_{12}\\left(2x+6\\right)+{\\mathrm{log}}_{12}\\left(x+2\\right)=2[\/latex]. Show the steps for solving.\r\n\r\n39. Use the quotient rule for logarithms to find all x<\/em>\u00a0values such that [latex]{\\mathrm{log}}_{6}\\left(x+2\\right)-{\\mathrm{log}}_{6}\\left(x - 3\\right)=1[\/latex]. Show the steps for solving.\r\n\r\n40. 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.\r\n\r\n41. Prove that [latex]{\\mathrm{log}}_{b}\\left(n\\right)=\\frac{1}{{\\mathrm{log}}_{n}\\left(b\\right)}[\/latex] for any positive integers b\u00a0<\/em>> 1 and n\u00a0<\/em>> 1.\r\n\r\n42.\u00a0Does [latex]{\\mathrm{log}}_{81}\\left(2401\\right)={\\mathrm{log}}_{3}\\left(7\\right)[\/latex]? Verify the claim algebraically.","rendered":"

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]?<\/p>\n

2.\u00a0What does the change-of-base formula do? Why is it useful when using a calculator?<\/p>\n

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.<\/p>\n

3.\u00a0[latex]{\\mathrm{log}}_{b}\\left(7x\\cdot 2y\\right)[\/latex]<\/p>\n

4.\u00a0[latex]\\mathrm{ln}\\left(3ab\\cdot 5c\\right)[\/latex]<\/p>\n

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

6.\u00a0[latex]{\\mathrm{log}}_{4}\\left(\\frac{\\text{ }\\frac{x}{z}\\text{ }}{w}\\right)[\/latex]<\/p>\n

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

8.\u00a0[latex]{\\mathrm{log}}_{2}\\left({y}^{x}\\right)[\/latex]<\/p>\n

For the following exercises, condense to a single logarithm if possible.<\/p>\n

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

10.\u00a0[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]<\/p>\n

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

12.\u00a0[latex]\\mathrm{ln}\\left(a\\right)-\\mathrm{ln}\\left(d\\right)-\\mathrm{ln}\\left(c\\right)[\/latex]<\/p>\n

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

14.\u00a0[latex]\\frac{1}{3}\\mathrm{ln}\\left(8\\right)[\/latex]<\/p>\n

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.<\/p>\n

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

16.\u00a0[latex]\\mathrm{ln}\\left(\\frac{{a}^{-2}}{{b}^{-4}{c}^{5}}\\right)[\/latex]<\/p>\n

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

18.\u00a0[latex]\\mathrm{ln}\\left(y\\sqrt{\\frac{y}{1-y}}\\right)[\/latex]<\/p>\n

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

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.<\/p>\n

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

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

22.\u00a0[latex]2\\mathrm{log}\\left(x\\right)+3\\mathrm{log}\\left(x+1\\right)[\/latex]<\/p>\n

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

24.\u00a0[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]<\/p>\n

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.<\/p>\n

25. [latex]{\\mathrm{log}}_{7}\\left(15\\right)[\/latex] to base\u00a0e<\/em><\/p>\n

26.\u00a0[latex]{\\mathrm{log}}_{14}\\left(55.875\\right)[\/latex] to base 10<\/p>\n

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<\/em>\u00a0and b<\/em>. Show the steps for solving.<\/p>\n

27. [latex]{\\mathrm{log}}_{11}\\left(5\\right)[\/latex]<\/p>\n

28.\u00a0[latex]{\\mathrm{log}}_{6}\\left(55\\right)[\/latex]<\/p>\n

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

For the following exercises, use properties of logarithms to evaluate without using a calculator.<\/p>\n

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

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

32.\u00a0[latex]2{\\mathrm{log}}_{9}\\left(3\\right)-4{\\mathrm{log}}_{9}\\left(3\\right)+{\\mathrm{log}}_{9}\\left(\\frac{1}{729}\\right)[\/latex]<\/p>\n

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.<\/p>\n

33. [latex]{\\mathrm{log}}_{3}\\left(22\\right)[\/latex]<\/p>\n

34.\u00a0[latex]{\\mathrm{log}}_{8}\\left(65\\right)[\/latex]<\/p>\n

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

36.\u00a0[latex]{\\mathrm{log}}_{4}\\left(\\frac{15}{2}\\right)[\/latex]<\/p>\n

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

38. Use the product rule for logarithms to find all x<\/em>\u00a0values such that [latex]{\\mathrm{log}}_{12}\\left(2x+6\\right)+{\\mathrm{log}}_{12}\\left(x+2\\right)=2[\/latex]. Show the steps for solving.<\/p>\n

39. Use the quotient rule for logarithms to find all x<\/em>\u00a0values such that [latex]{\\mathrm{log}}_{6}\\left(x+2\\right)-{\\mathrm{log}}_{6}\\left(x - 3\\right)=1[\/latex]. Show the steps for solving.<\/p>\n

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.<\/p>\n

41. Prove that [latex]{\\mathrm{log}}_{b}\\left(n\\right)=\\frac{1}{{\\mathrm{log}}_{n}\\left(b\\right)}[\/latex] for any positive integers b\u00a0<\/em>> 1 and n\u00a0<\/em>> 1.<\/p>\n

42.\u00a0Does [latex]{\\mathrm{log}}_{81}\\left(2401\\right)={\\mathrm{log}}_{3}\\left(7\\right)[\/latex]? Verify the claim algebraically.<\/p>\n\n\t\t\t

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