{"id":11197,"date":"2015-07-14T18:46:57","date_gmt":"2015-07-14T18:46:57","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&#038;p=11197"},"modified":"2018-06-28T14:52:29","modified_gmt":"2018-06-28T14:52:29","slug":"section-exercises-20","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/section-exercises-20\/","title":{"raw":"Section Exercises 5.4: Graphs of Logarithmic Functions","rendered":"Section Exercises 5.4: Graphs of Logarithmic Functions"},"content":{"raw":"1. The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?\r\n\r\n2.\u00a0What type(s) of translation(s), if any, affect the range of a logarithmic function?\r\n\r\n3. What type(s) of translation(s), if any, affect the domain of a logarithmic function?\r\n\r\n4.\u00a0Consider the general logarithmic function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Why can\u2019t <em>x<\/em>\u00a0be zero?\r\n\r\n5. Does the graph of a general logarithmic function have a horizontal asymptote? Explain.\r\n\r\nFor the following exercises, state the domain and range of the function.\r\n\r\n6. [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex]\r\n\r\n7. [latex]h\\left(x\\right)=\\mathrm{ln}\\left(\\frac{1}{2}-x\\right)[\/latex]\r\n\r\n8.\u00a0[latex]g\\left(x\\right)={\\mathrm{log}}_{5}\\left(2x+9\\right)-2[\/latex]\r\n\r\n9. [latex]h\\left(x\\right)=\\mathrm{ln}\\left(4x+17\\right)-5[\/latex]\r\n\r\n10.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(12 - 3x\\right)-3[\/latex]\r\n\r\nFor the following exercises, state the domain and the vertical asymptote of the function.\r\n\r\n11. [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x - 5\\right)[\/latex]\r\n\r\n12.\u00a0[latex]g\\left(x\\right)=\\mathrm{ln}\\left(3-x\\right)[\/latex]\r\n\r\n13. [latex]f\\left(x\\right)=\\mathrm{log}\\left(3x+1\\right)[\/latex]\r\n\r\n14.\u00a0[latex]f\\left(x\\right)=3\\mathrm{log}\\left(-x\\right)+2[\/latex]\r\n\r\n15. [latex]g\\left(x\\right)=-\\mathrm{ln}\\left(3x+9\\right)-7[\/latex]\r\n\r\nFor the following exercises, state the domain, vertical asymptote, and end behavior of the function.\r\n\r\n16. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(2-x\\right)[\/latex]\r\n\r\n17. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x-\\frac{3}{7}\\right)[\/latex]\r\n\r\n18.\u00a0[latex]h\\left(x\\right)=-\\mathrm{log}\\left(3x - 4\\right)+3[\/latex]\r\n\r\n19. [latex]g\\left(x\\right)=\\mathrm{ln}\\left(2x+6\\right)-5[\/latex]\r\n\r\n20.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(15 - 5x\\right)+6[\/latex]\r\n\r\nFor the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.\r\n\r\n21. [latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x - 1\\right)+1[\/latex]\r\n\r\n22.\u00a0[latex]f\\left(x\\right)=\\mathrm{log}\\left(5x+10\\right)+3[\/latex]\r\n\r\n23. [latex]g\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)-2[\/latex]\r\n\r\n24.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)-5[\/latex]\r\n\r\n25. [latex]h\\left(x\\right)=3\\mathrm{ln}\\left(x\\right)-9[\/latex]\r\n\r\nFor the following exercises, match each function in the graph below\u00a0with the letter corresponding to its graph.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005324\/CNX_PreCalc_Figure_04_04_201.jpg\" alt=\"Graph of five logarithmic functions.\" \/>\r\n\r\n26. [latex]d\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex]\r\n\r\n27. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]\r\n\r\n28. [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]\r\n\r\n29. [latex]h\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]\r\n\r\n30.\u00a0[latex]j\\left(x\\right)={\\mathrm{log}}_{25}\\left(x\\right)[\/latex]\r\n\r\nFor the following exercises, match each function in the figure below\u00a0with the letter corresponding to its graph.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005324\/CNX_PreCalc_Figure_04_04_202.jpg\" alt=\"Graph of three logarithmic functions.\" \/>\r\n\r\n31.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{3}}\\left(x\\right)[\/latex]\r\n\r\n32. [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]\r\n\r\n33. [latex]h\\left(x\\right)={\\mathrm{log}}_{\\frac{3}{4}}\\left(x\\right)[\/latex]\r\n\r\nFor the following exercises, sketch the graphs of each pair of functions on the same axis.\r\n\r\n34. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)={10}^{x}[\/latex]\r\n\r\n35. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)[\/latex]\r\n\r\n36. [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]\r\n\r\n37. [latex]f\\left(x\\right)={e}^{x}[\/latex] and [latex]g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]\r\n\r\nFor the following exercises, match each function in the graph below\u00a0with the letter corresponding to its graph.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005325\/CNX_PreCalc_Figure_04_04_207.jpg\" alt=\"Graph of three logarithmic functions.\" \/>\r\n38. [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(-x+2\\right)[\/latex]\r\n\r\n39. [latex]g\\left(x\\right)=-{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]\r\n\r\n40. [latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]\r\n\r\nFor the following exercises, sketch the graph of the indicated function.\r\n\r\n41. [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)[\/latex]\r\n\r\n42. [latex]f\\left(x\\right)=2\\mathrm{log}\\left(x\\right)[\/latex]\r\n\r\n43. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)[\/latex]\r\n\r\n44. [latex]g\\left(x\\right)=\\mathrm{log}\\left(4x+16\\right)+4[\/latex]\r\n\r\n45. [latex]g\\left(x\\right)=\\mathrm{log}\\left(6 - 3x\\right)+1[\/latex]\r\n\r\n46. [latex]h\\left(x\\right)=-\\frac{1}{2}\\mathrm{ln}\\left(x+1\\right)-3[\/latex]\r\n\r\nFor the following exercises, write a logarithmic equation corresponding to the graph shown.\r\n\r\n47. Use [latex]y={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] as the parent function.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005326\/CNX_PreCalc_Figure_04_04_214.jpg\" alt=\"The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.\" \/>\r\n\r\n48.\u00a0Use [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] as the parent function.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005326\/CNX_PreCalc_Figure_04_04_215.jpg\" alt=\"The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.\" \/>\r\n\r\n49. Use [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] as the parent function.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005326\/CNX_PreCalc_Figure_04_04_216.jpg\" alt=\"The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.\" \/>\r\n\r\n50.\u00a0Use [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex] as the parent function.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005326\/CNX_PreCalc_Figure_04_04_217.jpg\" alt=\"The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.\" \/>\r\n\r\nFor the following exercises, use a graphing calculator to find approximate solutions to each equation.\r\n\r\n51. [latex]\\mathrm{log}\\left(x - 1\\right)+2=\\mathrm{ln}\\left(x - 1\\right)+2[\/latex]\r\n\r\n52.\u00a0[latex]\\mathrm{log}\\left(2x - 3\\right)+2=-\\mathrm{log}\\left(2x - 3\\right)+5[\/latex]\r\n\r\n53. [latex]\\mathrm{ln}\\left(x - 2\\right)=-\\mathrm{ln}\\left(x+1\\right)[\/latex]\r\n\r\n54. [latex]2\\mathrm{ln}\\left(5x+1\\right)=\\frac{1}{2}\\mathrm{ln}\\left(-5x\\right)+1[\/latex]\r\n\r\n55. [latex]\\frac{1}{3}\\mathrm{log}\\left(1-x\\right)=\\mathrm{log}\\left(x+1\\right)+\\frac{1}{3}[\/latex]\r\n\r\n56. Let <em>b<\/em>\u00a0be any positive real number such that [latex]b\\ne 1[\/latex]. What must [latex]{\\mathrm{log}}_{b}1[\/latex] be equal to? Verify the result.\r\n\r\n57. Explore and discuss the graphs of [latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right)[\/latex]. Make a conjecture based on the result.\r\n\r\n58.\u00a0Prove the conjecture made in the previous exercise.\r\n\r\n59. What is the domain of the function [latex]f\\left(x\\right)=\\mathrm{ln}\\left(\\frac{x+2}{x - 4}\\right)[\/latex]? Discuss the result.\r\n\r\n60.\u00a0Use properties of exponents to find the x-intercepts of the function [latex]f\\left(x\\right)=\\mathrm{log}\\left({x}^{2}+4x+4\\right)[\/latex] algebraically. Show the steps for solving, and then verify the result by graphing the function.","rendered":"<p>1. The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?<\/p>\n<p>2.\u00a0What type(s) of translation(s), if any, affect the range of a logarithmic function?<\/p>\n<p>3. What type(s) of translation(s), if any, affect the domain of a logarithmic function?<\/p>\n<p>4.\u00a0Consider the general logarithmic function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Why can\u2019t <em>x<\/em>\u00a0be zero?<\/p>\n<p>5. Does the graph of a general logarithmic function have a horizontal asymptote? Explain.<\/p>\n<p>For the following exercises, state the domain and range of the function.<\/p>\n<p>6. [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex]<\/p>\n<p>7. [latex]h\\left(x\\right)=\\mathrm{ln}\\left(\\frac{1}{2}-x\\right)[\/latex]<\/p>\n<p>8.\u00a0[latex]g\\left(x\\right)={\\mathrm{log}}_{5}\\left(2x+9\\right)-2[\/latex]<\/p>\n<p>9. [latex]h\\left(x\\right)=\\mathrm{ln}\\left(4x+17\\right)-5[\/latex]<\/p>\n<p>10.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(12 - 3x\\right)-3[\/latex]<\/p>\n<p>For the following exercises, state the domain and the vertical asymptote of the function.<\/p>\n<p>11. [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x - 5\\right)[\/latex]<\/p>\n<p>12.\u00a0[latex]g\\left(x\\right)=\\mathrm{ln}\\left(3-x\\right)[\/latex]<\/p>\n<p>13. [latex]f\\left(x\\right)=\\mathrm{log}\\left(3x+1\\right)[\/latex]<\/p>\n<p>14.\u00a0[latex]f\\left(x\\right)=3\\mathrm{log}\\left(-x\\right)+2[\/latex]<\/p>\n<p>15. [latex]g\\left(x\\right)=-\\mathrm{ln}\\left(3x+9\\right)-7[\/latex]<\/p>\n<p>For the following exercises, state the domain, vertical asymptote, and end behavior of the function.<\/p>\n<p>16. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(2-x\\right)[\/latex]<\/p>\n<p>17. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x-\\frac{3}{7}\\right)[\/latex]<\/p>\n<p>18.\u00a0[latex]h\\left(x\\right)=-\\mathrm{log}\\left(3x - 4\\right)+3[\/latex]<\/p>\n<p>19. [latex]g\\left(x\\right)=\\mathrm{ln}\\left(2x+6\\right)-5[\/latex]<\/p>\n<p>20.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(15 - 5x\\right)+6[\/latex]<\/p>\n<p>For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.<\/p>\n<p>21. [latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x - 1\\right)+1[\/latex]<\/p>\n<p>22.\u00a0[latex]f\\left(x\\right)=\\mathrm{log}\\left(5x+10\\right)+3[\/latex]<\/p>\n<p>23. [latex]g\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)-2[\/latex]<\/p>\n<p>24.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)-5[\/latex]<\/p>\n<p>25. [latex]h\\left(x\\right)=3\\mathrm{ln}\\left(x\\right)-9[\/latex]<\/p>\n<p>For the following exercises, match each function in the graph below\u00a0with the letter corresponding to its graph.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005324\/CNX_PreCalc_Figure_04_04_201.jpg\" alt=\"Graph of five logarithmic functions.\" \/><\/p>\n<p>26. [latex]d\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\n<p>27. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<p>28. [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\n<p>29. [latex]h\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]<\/p>\n<p>30.\u00a0[latex]j\\left(x\\right)={\\mathrm{log}}_{25}\\left(x\\right)[\/latex]<\/p>\n<p>For the following exercises, match each function in the figure below\u00a0with the letter corresponding to its graph.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005324\/CNX_PreCalc_Figure_04_04_202.jpg\" alt=\"Graph of three logarithmic functions.\" \/><\/p>\n<p>31.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{3}}\\left(x\\right)[\/latex]<\/p>\n<p>32. [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\n<p>33. [latex]h\\left(x\\right)={\\mathrm{log}}_{\\frac{3}{4}}\\left(x\\right)[\/latex]<\/p>\n<p>For the following exercises, sketch the graphs of each pair of functions on the same axis.<\/p>\n<p>34. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)={10}^{x}[\/latex]<\/p>\n<p>35. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)[\/latex]<\/p>\n<p>36. [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<p>37. [latex]f\\left(x\\right)={e}^{x}[\/latex] and [latex]g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<p>For the following exercises, match each function in the graph below\u00a0with the letter corresponding to its graph.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005325\/CNX_PreCalc_Figure_04_04_207.jpg\" alt=\"Graph of three logarithmic functions.\" \/><br \/>\n38. [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(-x+2\\right)[\/latex]<\/p>\n<p>39. [latex]g\\left(x\\right)=-{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\n<p>40. [latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\n<p>For the following exercises, sketch the graph of the indicated function.<\/p>\n<p>41. [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)[\/latex]<\/p>\n<p>42. [latex]f\\left(x\\right)=2\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\n<p>43. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)[\/latex]<\/p>\n<p>44. [latex]g\\left(x\\right)=\\mathrm{log}\\left(4x+16\\right)+4[\/latex]<\/p>\n<p>45. [latex]g\\left(x\\right)=\\mathrm{log}\\left(6 - 3x\\right)+1[\/latex]<\/p>\n<p>46. [latex]h\\left(x\\right)=-\\frac{1}{2}\\mathrm{ln}\\left(x+1\\right)-3[\/latex]<\/p>\n<p>For the following exercises, write a logarithmic equation corresponding to the graph shown.<\/p>\n<p>47. Use [latex]y={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] as the parent function.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005326\/CNX_PreCalc_Figure_04_04_214.jpg\" alt=\"The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.\" \/><\/p>\n<p>48.\u00a0Use [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] as the parent function.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005326\/CNX_PreCalc_Figure_04_04_215.jpg\" alt=\"The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.\" \/><\/p>\n<p>49. Use [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] as the parent function.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005326\/CNX_PreCalc_Figure_04_04_216.jpg\" alt=\"The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.\" \/><\/p>\n<p>50.\u00a0Use [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex] as the parent function.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005326\/CNX_PreCalc_Figure_04_04_217.jpg\" alt=\"The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.\" \/><\/p>\n<p>For the following exercises, use a graphing calculator to find approximate solutions to each equation.<\/p>\n<p>51. [latex]\\mathrm{log}\\left(x - 1\\right)+2=\\mathrm{ln}\\left(x - 1\\right)+2[\/latex]<\/p>\n<p>52.\u00a0[latex]\\mathrm{log}\\left(2x - 3\\right)+2=-\\mathrm{log}\\left(2x - 3\\right)+5[\/latex]<\/p>\n<p>53. [latex]\\mathrm{ln}\\left(x - 2\\right)=-\\mathrm{ln}\\left(x+1\\right)[\/latex]<\/p>\n<p>54. [latex]2\\mathrm{ln}\\left(5x+1\\right)=\\frac{1}{2}\\mathrm{ln}\\left(-5x\\right)+1[\/latex]<\/p>\n<p>55. [latex]\\frac{1}{3}\\mathrm{log}\\left(1-x\\right)=\\mathrm{log}\\left(x+1\\right)+\\frac{1}{3}[\/latex]<\/p>\n<p>56. Let <em>b<\/em>\u00a0be any positive real number such that [latex]b\\ne 1[\/latex]. What must [latex]{\\mathrm{log}}_{b}1[\/latex] be equal to? Verify the result.<\/p>\n<p>57. Explore and discuss the graphs of [latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right)[\/latex]. Make a conjecture based on the result.<\/p>\n<p>58.\u00a0Prove the conjecture made in the previous exercise.<\/p>\n<p>59. What is the domain of the function [latex]f\\left(x\\right)=\\mathrm{ln}\\left(\\frac{x+2}{x - 4}\\right)[\/latex]? Discuss the result.<\/p>\n<p>60.\u00a0Use properties of exponents to find the x-intercepts of the function [latex]f\\left(x\\right)=\\mathrm{log}\\left({x}^{2}+4x+4\\right)[\/latex] algebraically. Show the steps for solving, and then verify the result by graphing the function.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-11197\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":276,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-11197","chapter","type-chapter","status-publish","hentry"],"part":11188,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/11197","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/11197\/revisions"}],"predecessor-version":[{"id":15148,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/11197\/revisions\/15148"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/parts\/11188"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapters\/11197\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/media?parent=11197"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/pressbooks\/v2\/chapter-type?post=11197"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/contributor?post=11197"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/wp-json\/wp\/v2\/license?post=11197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}