{"id":2312,"date":"2019-05-10T14:10:58","date_gmt":"2019-05-10T14:10:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2312"},"modified":"2019-05-10T14:10:58","modified_gmt":"2019-05-10T14:10:58","slug":"2-6-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/2-6-section-exercises\/","title":{"raw":"2.6 Section Exercises","rendered":"2.6 Section Exercises"},"content":{"raw":"
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2.6 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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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>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135582193\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135582193\"]\r\n

Since the functions are inverses, their graphs are mirror images about the line[latex]\\text{ }y=x.\\text{ }[\/latex]So for every point[latex]\\text{ }\\left(a,b\\right)\\text{ }[\/latex]on the graph of a logarithmic function, there is a corresponding point[latex]\\text{ }\\left(b,a\\right)\\text{ }[\/latex]on the graph of its inverse exponential function.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. What type(s) of translation(s), if any, affect the range of a logarithmic function?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. What type(s) of translation(s), if any, affect the domain of a logarithmic function?<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n4. Shifting the function right or left and reflecting the function about the y-axis will affect its domain.\r\n\r\n<\/div>\r\n<\/div>\r\n
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5. Consider the general logarithmic function[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right).\\text{ }[\/latex]Why can\u2019t[latex]\\text{ }x\\text{ }[\/latex]be zero?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6. Does the graph of a general logarithmic function have a horizontal asymptote? Explain.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137459912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137459912\"]\r\n

No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Algebraic<\/h4>\r\n

For the following exercises, state the domain and range of the function.<\/p>\r\n\r\n

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\r\n\r\n7. [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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8. [latex]h\\left(x\\right)=\\mathrm{ln}\\left(\\frac{1}{2}-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135257219\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135257219\"]\r\n

Domain:[latex]\\text{ }\\left(-\\infty ,\\frac{1}{2}\\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [latex]g\\left(x\\right)={\\mathrm{log}}_{5}\\left(2x+9\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. [latex]h\\left(x\\right)=\\mathrm{ln}\\left(4x+17\\right)-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137641046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137641046\"]\r\n

Domain:[latex]\\text{ }\\left(-\\frac{17}{4},\\infty \\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(12-3x\\right)-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, state the domain and the vertical asymptote of the function.<\/p>\r\n\r\n

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12. [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-5\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137771436\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137771436\"]\r\n

Domain:[latex]\\text{ }\\left(5,\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [latex]\\text{ }g\\left(x\\right)=\\mathrm{ln}\\left(3-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(3x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135256140\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135256140\"]\r\n

Domain:[latex]\\text{ }\\left(-\\frac{1}{3},\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=-\\frac{1}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [latex]\\text{ }f\\left(x\\right)=3\\mathrm{log}\\left(-x\\right)+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. [latex]\\text{ }g\\left(x\\right)=-\\mathrm{ln}\\left(3x+9\\right)-7[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135532384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135532384\"]\r\n

Domain:[latex]\\text{ }\\left(-3,\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=-3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, state the domain, vertical asymptote, and end behavior of the function.<\/p>\r\n\r\n

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17. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(2-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x-\\frac{3}{7}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137715430\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137715430\"]\r\n

Domain: [latex]\\left(\\frac{3}{7},\\infty \\right)[\/latex];<\/p>\r\n\r\n

[\/hidden-answer]<\/div>\r\n19. Vertical asymptote: [latex]x=\\frac{3}{7}[\/latex]; End behavior: as [latex]x\\to {\\left(\\frac{3}{7}\\right)}^{+},f\\left(x\\right)\\to -\\infty [\/latex] and as [latex]x\\to \\infty ,f\\left(x\\right)\\to \\infty [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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20. [latex]h\\left(x\\right)=-\\mathrm{log}\\left(3x-4\\right)+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [latex]g\\left(x\\right)=\\mathrm{ln}\\left(2x+6\\right)-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135358044\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135358044\"]\r\n

Domain: [latex]\\left(-3,\\infty \\right)[\/latex]; Vertical asymptote: [latex]x=-3[\/latex];<\/p>\r\n\r\n

[\/hidden-answer]<\/div>\r\n22. End behavior: as [latex]x\\to -{3}^{+}[\/latex], [latex]f\\left(x\\right)\\to -\\infty [\/latex] and as [latex]x\\to \\infty [\/latex], [latex]f\\left(x\\right)\\to \\infty [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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23. [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(15-5x\\right)+6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, state the domain, range, and x<\/em>- and y<\/em>-intercepts, if they exist. If they do not exist, write DNE.<\/p>\r\n\r\n

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24. [latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x-1\\right)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165132983045\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132983045\"]\r\n

Domain:[latex]\\text{ }\\left(1,\\infty \\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=1;\\text{ }[\/latex]x<\/em>-intercept:[latex]\\text{ }\\left(\\frac{5}{4},0\\right);\\text{ }[\/latex]y<\/em>-intercept: DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. [latex]f\\left(x\\right)=\\mathrm{log}\\left(5x+10\\right)+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. [latex]g\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137426893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137426893\"]\r\n

Domain:[latex]\\text{ }\\left(-\\infty ,0\\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=0;\\text{ }[\/latex]x<\/em>-intercept:[latex]\\text{ }\\left(-{e}^{2},0\\right);\\text{ }[\/latex]y<\/em>-intercept: DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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27. [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n28. [latex]h\\left(x\\right)=3\\mathrm{ln}\\left(x\\right)-9[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135186502\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135186502\"]\r\n

Domain:[latex]\\text{ }\\left(0,\\infty \\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right);\\text{ }[\/latex] Vertical asymptote: [latex]\\text{ }x=0;\\text{ }[\/latex]x<\/em>-intercept:[latex]\\text{ }\\left({e}^{3},0\\right);\\text{ }[\/latex]y<\/em>-intercept: DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Graphical<\/h4>\r\n

For the following exercises, match each function in (Figure)<\/a> with the letter corresponding to its graph.<\/p>\r\n\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 17.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n
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29. [latex]d\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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30. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134255558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134255558\"]\r\n

B<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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31. [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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32. [latex]h\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"328543\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"328543\"]C[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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33. [latex]j\\left(x\\right)={\\mathrm{log}}_{25}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises, match each function in (Figure)<\/a> with the letter corresponding to its graph.\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"342\"]\"Graph Figure 18.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n
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34. [latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{3}}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135206082\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135206082\"]B[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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35. [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36. [latex]h\\left(x\\right)={\\mathrm{log}}_{\\frac{3}{4}}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137637948\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137637948\"]\r\n

C<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, sketch the graphs of each pair of functions on the same axis.<\/p>\r\n\r\n

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37. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)={10}^{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"449456\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"449456\"]\"Graph<\/div>\r\n
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39. [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. [latex]f\\left(x\\right)={e}^{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135496605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135496605\"]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n

For the following exercises, match each function in (Figure)<\/a> with the letter corresponding to its graph.<\/p>\r\n\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"374\"]\"Graph Figure 19.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n
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41. [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(-x+2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. [latex]g\\left(x\\right)=-{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n[reveal-answer q=\"677321\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"677321\"]C[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n
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43. [latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises, sketch the graph of the indicated function.\r\n

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44. [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"893879\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"893879\"]\"Graph<\/span><\/div>\r\n
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45. [latex]\\text{ }f\\left(x\\right)=2\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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46. [latex]\\text{ }f\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134032402\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134032402\"]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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47. [latex]g\\left(x\\right)=\\mathrm{log}\\left(4x+16\\right)+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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48. [latex]g\\left(x\\right)=\\mathrm{log}\\left(6-3x\\right)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137705397\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137705397\"]\"Graph[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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49. [latex]h\\left(x\\right)=-\\frac{1}{2}\\mathrm{ln}\\left(x+1\\right)-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, write a logarithmic equation corresponding to the graph shown.<\/p>\r\n\r\n

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50. Use[latex]\\text{ }y={\\mathrm{log}}_{2}\\left(x\\right)\\text{ }[\/latex]as the parent function.<\/p>\r\n\"The<\/span>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134360851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134360851\"]\r\n

[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{2}\\left(-\\left(x-1\\right)\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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51. Use[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\text{ }[\/latex]as the parent function.<\/p>\r\n\"The<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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52. Use[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\text{ }[\/latex]as the parent function.<\/p>\r\n\"The<\/span>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134086084\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134086084\"]\r\n

[latex]f\\left(x\\right)=3{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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53. Use[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)\\text{ }[\/latex]as the parent function.<\/p>\r\n\"The\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Technology<\/h4>\r\n

For the following exercises, use a graphing calculator to find approximate solutions to each equation.<\/p>\r\n\r\n

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54. [latex]\\mathrm{log}\\left(x-1\\right)+2=\\mathrm{ln}\\left(x-1\\right)+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137674291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137674291\"]\r\n

[latex]x=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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55. [latex]\\mathrm{log}\\left(2x-3\\right)+2=-\\mathrm{log}\\left(2x-3\\right)+5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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56. [latex]\\mathrm{ln}\\left(x-2\\right)=-\\mathrm{ln}\\left(x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135337765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135337765\"]\r\n

[latex]x\\approx \\text{2}\\text{.303}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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57. [latex]2\\mathrm{ln}\\left(5x+1\\right)=\\frac{1}{2}\\mathrm{ln}\\left(-5x\\right)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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58. [latex]\\frac{1}{3}\\mathrm{log}\\left(1-x\\right)=\\mathrm{log}\\left(x+1\\right)+\\frac{1}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135174505\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135174505\"]\r\n

[latex]x\\approx -0.472[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Extensions<\/h4>\r\n
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59. Let[latex]\\text{ }b\\text{ }[\/latex]be any positive real number such that[latex]\\text{ }b\\ne 1.\\text{ }[\/latex]What must[latex]\\text{ }{\\mathrm{log}}_{b}1\\text{ }[\/latex]be equal to? Verify the result.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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60. Explore and discuss the graphs of[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right).\\text{ }[\/latex]Make a conjecture based on the result.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134151930\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134151930\"]\r\n

The graphs of[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right)\\text{ }[\/latex]appear to be the same; Conjecture: for any positive base[latex]\\text{ }b\\ne 1,[\/latex][latex]\\text{ }{\\mathrm{log}}_{b}\\left(x\\right)=-{\\mathrm{log}}_{\\frac{1}{b}}\\left(x\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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61. Prove the conjecture made in the previous exercise.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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62. What is the domain of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{ln}\\left(\\frac{x+2}{x-4}\\right)?\\text{ }[\/latex]Discuss the result.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135593381\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135593381\"]\r\n

Recall that the argument of a logarithmic function must be positive, so we determine where[latex]\\text{ }\\frac{x+2}{x-4}>0\\text{ }[\/latex]. From the graph of the function[latex]\\text{ }f\\left(x\\right)=\\frac{x+2}{x-4},[\/latex] note that the graph lies above the x<\/em>-axis on the interval[latex]\\text{ }\\left(-\\infty ,-2\\right)\\text{ }[\/latex]and again to the right of the vertical asymptote, that is[latex]\\text{ }\\left(4,\\infty \\right).\\text{ }[\/latex]Therefore, the domain is[latex]\\text{ }\\left(-\\infty ,-2\\right)\\cup \\left(4,\\infty \\right).[\/latex]<\/p>\r\n\"\"<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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63. Use properties of exponents to find the x<\/em>-intercepts of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left({x}^{2}+4x+4\\right)\\text{ }[\/latex]algebraically. Show the steps for solving, and then verify the result by graphing the function.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

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2.6 Section Exercises<\/h3>\n
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Verbal<\/h4>\n
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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<\/div>\n

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Show Solution<\/span><\/p>\n
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Since the functions are inverses, their graphs are mirror images about the line[latex]\\text{ }y=x.\\text{ }[\/latex]So for every point[latex]\\text{ }\\left(a,b\\right)\\text{ }[\/latex]on the graph of a logarithmic function, there is a corresponding point[latex]\\text{ }\\left(b,a\\right)\\text{ }[\/latex]on the graph of its inverse exponential function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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2. What type(s) of translation(s), if any, affect the range of a logarithmic function?<\/p>\n<\/div>\n<\/div>\n

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3. What type(s) of translation(s), if any, affect the domain of a logarithmic function?<\/p>\n<\/div>\n

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4. Shifting the function right or left and reflecting the function about the y-axis will affect its domain.<\/p>\n<\/div>\n<\/div>\n

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5. Consider the general logarithmic function[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right).\\text{ }[\/latex]Why can\u2019t[latex]\\text{ }x\\text{ }[\/latex]be zero?<\/p>\n<\/div>\n<\/div>\n

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6. Does the graph of a general logarithmic function have a horizontal asymptote? Explain.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Algebraic<\/h4>\n

For the following exercises, state the domain and range of the function.<\/p>\n

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7. [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x+4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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8. [latex]h\\left(x\\right)=\\mathrm{ln}\\left(\\frac{1}{2}-x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Domain:[latex]\\text{ }\\left(-\\infty ,\\frac{1}{2}\\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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9. [latex]g\\left(x\\right)={\\mathrm{log}}_{5}\\left(2x+9\\right)-2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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10. [latex]h\\left(x\\right)=\\mathrm{ln}\\left(4x+17\\right)-5[\/latex]<\/p>\n<\/div>\n

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Domain:[latex]\\text{ }\\left(-\\frac{17}{4},\\infty \\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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11. [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(12-3x\\right)-3[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, state the domain and the vertical asymptote of the function.<\/p>\n

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12. [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-5\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Domain:[latex]\\text{ }\\left(5,\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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13. [latex]\\text{ }g\\left(x\\right)=\\mathrm{ln}\\left(3-x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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14. [latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(3x+1\\right)[\/latex]<\/p>\n<\/div>\n

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Domain:[latex]\\text{ }\\left(-\\frac{1}{3},\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=-\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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15. [latex]\\text{ }f\\left(x\\right)=3\\mathrm{log}\\left(-x\\right)+2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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16. [latex]\\text{ }g\\left(x\\right)=-\\mathrm{ln}\\left(3x+9\\right)-7[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Domain:[latex]\\text{ }\\left(-3,\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, state the domain, vertical asymptote, and end behavior of the function.<\/p>\n

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17. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(2-x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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18. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x-\\frac{3}{7}\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Domain: [latex]\\left(\\frac{3}{7},\\infty \\right)[\/latex];<\/p>\n

<\/div>\n<\/div>\n<\/div>\n

19. Vertical asymptote: [latex]x=\\frac{3}{7}[\/latex]; End behavior: as [latex]x\\to {\\left(\\frac{3}{7}\\right)}^{+},f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to \\infty ,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n

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20. [latex]h\\left(x\\right)=-\\mathrm{log}\\left(3x-4\\right)+3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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21. [latex]g\\left(x\\right)=\\mathrm{ln}\\left(2x+6\\right)-5[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Domain: [latex]\\left(-3,\\infty \\right)[\/latex]; Vertical asymptote: [latex]x=-3[\/latex];<\/p>\n

<\/div>\n<\/div>\n<\/div>\n

22. End behavior: as [latex]x\\to -{3}^{+}[\/latex], [latex]f\\left(x\\right)\\to -\\infty[\/latex] and as [latex]x\\to \\infty[\/latex], [latex]f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/div>\n<\/div>\n

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23. [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(15-5x\\right)+6[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, state the domain, range, and x<\/em>– and y<\/em>-intercepts, if they exist. If they do not exist, write DNE.<\/p>\n

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24. [latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x-1\\right)+1[\/latex]<\/p>\n<\/div>\n

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Domain:[latex]\\text{ }\\left(1,\\infty \\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=1;\\text{ }[\/latex]x<\/em>-intercept:[latex]\\text{ }\\left(\\frac{5}{4},0\\right);\\text{ }[\/latex]y<\/em>-intercept: DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25. [latex]f\\left(x\\right)=\\mathrm{log}\\left(5x+10\\right)+3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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26. [latex]g\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)-2[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Domain:[latex]\\text{ }\\left(-\\infty ,0\\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right);\\text{ }[\/latex]Vertical asymptote:[latex]\\text{ }x=0;\\text{ }[\/latex]x<\/em>-intercept:[latex]\\text{ }\\left(-{e}^{2},0\\right);\\text{ }[\/latex]y<\/em>-intercept: DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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27. [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)-5[\/latex]<\/p>\n<\/div>\n<\/div>\n

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28. [latex]h\\left(x\\right)=3\\mathrm{ln}\\left(x\\right)-9[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Domain:[latex]\\text{ }\\left(0,\\infty \\right);\\text{ }[\/latex]Range:[latex]\\text{ }\\left(-\\infty ,\\infty \\right);\\text{ }[\/latex] Vertical asymptote: [latex]\\text{ }x=0;\\text{ }[\/latex]x<\/em>-intercept:[latex]\\text{ }\\left({e}^{3},0\\right);\\text{ }[\/latex]y<\/em>-intercept: DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Graphical<\/h4>\n

For the following exercises, match each function in (Figure)<\/a> with the letter corresponding to its graph.<\/p>\n

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\"Graph<\/p>\n

Figure 17.<\/strong><\/p>\n<\/div>\n<\/div>\n

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29. [latex]d\\left(x\\right)=\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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30. [latex]f\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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B<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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31. [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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32. [latex]h\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Answer<\/span><\/p>\n
C<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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33. [latex]j\\left(x\\right)={\\mathrm{log}}_{25}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, match each function in (Figure)<\/a> with the letter corresponding to its graph.<\/p>\n

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\"Graph<\/p>\n

Figure 18.<\/strong><\/p>\n<\/div>\n<\/div>\n

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34. [latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{3}}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
B<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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35. [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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36. [latex]h\\left(x\\right)={\\mathrm{log}}_{\\frac{3}{4}}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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C<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, sketch the graphs of each pair of functions on the same axis.<\/p>\n

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37. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)={10}^{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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38. [latex]f\\left(x\\right)=\\mathrm{log}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Answer<\/span><\/p>\n
\"Graph<\/div>\n
<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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39. [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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40. [latex]f\\left(x\\right)={e}^{x}\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=\\mathrm{ln}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, match each function in (Figure)<\/a> with the letter corresponding to its graph.<\/p>\n

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\"Graph<\/p>\n

Figure 19.<\/strong><\/p>\n<\/div>\n<\/div>\n

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41. [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(-x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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42. [latex]g\\left(x\\right)=-{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n

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Show Answer<\/span><\/p>\n
C<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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43. [latex]h\\left(x\\right)={\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, sketch the graph of the indicated function.<\/p>\n

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44. [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n

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Show Answer<\/span><\/p>\n
\"Graph<\/span><\/div>\n
<\/div>\n<\/div>\n

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45. [latex]\\text{ }f\\left(x\\right)=2\\mathrm{log}\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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46. [latex]\\text{ }f\\left(x\\right)=\\mathrm{ln}\\left(-x\\right)[\/latex]<\/p>\n<\/div>\n

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\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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47. [latex]g\\left(x\\right)=\\mathrm{log}\\left(4x+16\\right)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n

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48. [latex]g\\left(x\\right)=\\mathrm{log}\\left(6-3x\\right)+1[\/latex]<\/p>\n<\/div>\n

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\"Graph<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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49. [latex]h\\left(x\\right)=-\\frac{1}{2}\\mathrm{ln}\\left(x+1\\right)-3[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, write a logarithmic equation corresponding to the graph shown.<\/p>\n

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50. Use[latex]\\text{ }y={\\mathrm{log}}_{2}\\left(x\\right)\\text{ }[\/latex]as the parent function.<\/p>\n

\"The<\/span><\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{2}\\left(-\\left(x-1\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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51. Use[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)\\text{ }[\/latex]as the parent function.<\/p>\n

\"The<\/span><\/p>\n<\/div>\n<\/div>\n

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52. Use[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)\\text{ }[\/latex]as the parent function.<\/p>\n

\"The<\/span><\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)=3{\\mathrm{log}}_{4}\\left(x+2\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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53. Use[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)\\text{ }[\/latex]as the parent function.<\/p>\n

\"The<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Technology<\/h4>\n

For the following exercises, use a graphing calculator to find approximate solutions to each equation.<\/p>\n

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54. [latex]\\mathrm{log}\\left(x-1\\right)+2=\\mathrm{ln}\\left(x-1\\right)+2[\/latex]<\/p>\n<\/div>\n

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[latex]x=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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55. [latex]\\mathrm{log}\\left(2x-3\\right)+2=-\\mathrm{log}\\left(2x-3\\right)+5[\/latex]<\/p>\n<\/div>\n<\/div>\n

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56. [latex]\\mathrm{ln}\\left(x-2\\right)=-\\mathrm{ln}\\left(x+1\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]x\\approx \\text{2}\\text{.303}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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57. [latex]2\\mathrm{ln}\\left(5x+1\\right)=\\frac{1}{2}\\mathrm{ln}\\left(-5x\\right)+1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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58. [latex]\\frac{1}{3}\\mathrm{log}\\left(1-x\\right)=\\mathrm{log}\\left(x+1\\right)+\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x\\approx -0.472[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Extensions<\/h4>\n
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59. Let[latex]\\text{ }b\\text{ }[\/latex]be any positive real number such that[latex]\\text{ }b\\ne 1.\\text{ }[\/latex]What must[latex]\\text{ }{\\mathrm{log}}_{b}1\\text{ }[\/latex]be equal to? Verify the result.<\/p>\n<\/div>\n<\/div>\n

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60. Explore and discuss the graphs of[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right).\\text{ }[\/latex]Make a conjecture based on the result.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The graphs of[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{2}}\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }g\\left(x\\right)=-{\\mathrm{log}}_{2}\\left(x\\right)\\text{ }[\/latex]appear to be the same; Conjecture: for any positive base[latex]\\text{ }b\\ne 1,[\/latex][latex]\\text{ }{\\mathrm{log}}_{b}\\left(x\\right)=-{\\mathrm{log}}_{\\frac{1}{b}}\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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61. Prove the conjecture made in the previous exercise.<\/p>\n<\/div>\n<\/div>\n

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62. What is the domain of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{ln}\\left(\\frac{x+2}{x-4}\\right)?\\text{ }[\/latex]Discuss the result.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Recall that the argument of a logarithmic function must be positive, so we determine where[latex]\\text{ }\\frac{x+2}{x-4}>0\\text{ }[\/latex]. From the graph of the function[latex]\\text{ }f\\left(x\\right)=\\frac{x+2}{x-4},[\/latex] note that the graph lies above the x<\/em>-axis on the interval[latex]\\text{ }\\left(-\\infty ,-2\\right)\\text{ }[\/latex]and again to the right of the vertical asymptote, that is[latex]\\text{ }\\left(4,\\infty \\right).\\text{ }[\/latex]Therefore, the domain is[latex]\\text{ }\\left(-\\infty ,-2\\right)\\cup \\left(4,\\infty \\right).[\/latex]<\/p>\n

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63. Use properties of exponents to find the x<\/em>-intercepts of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left({x}^{2}+4x+4\\right)\\text{ }[\/latex]algebraically. Show the steps for solving, and then verify the result by graphing the function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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