{"id":2276,"date":"2019-05-09T14:16:31","date_gmt":"2019-05-09T14:16:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2276"},"modified":"2019-05-09T14:16:45","modified_gmt":"2019-05-09T14:16:45","slug":"2-5-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/2-5-section-exercises\/","title":{"raw":"2.5 Section Exercises","rendered":"2.5 Section Exercises"},"content":{"raw":"
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2.5 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. How can an exponential equation be solved?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137643496\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137643496\"]\r\n

Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. When does an extraneous solution occur? How can an extraneous solution be recognized?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135203846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135203846\"]\r\n

The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.<\/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, use like bases to solve the exponential equation.<\/p>\r\n\r\n

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4. [latex]{4}^{-3v-2}={4}^{-v}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5. [latex]64\\cdot {4}^{3x}=16[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137409289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137409289\"]\r\n

[latex]x=-\\frac{1}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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6. [latex]{3}^{2x+1}\\cdot {3}^{x}=243[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7. [latex]{2}^{-3n}\\cdot \\frac{1}{4}={2}^{n+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135187451\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135187451\"]\r\n

[latex]n=-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8. [latex]625\\cdot {5}^{3x+3}=125[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [latex]\\frac{{36}^{3b}}{{36}^{2b}}={216}^{2-b}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137418206\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137418206\"]\r\n

[latex]b=\\frac{6}{5}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n10. [latex]{\\left(\\frac{1}{64}\\right)}^{3n}\\cdot 8={2}^{6}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use logarithms to solve.<\/p>\r\n\r\n

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11. [latex]{9}^{x-10}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137425852\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137425852\"]\r\n

[latex]x=10[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12. [latex]2{e}^{6x}=13[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [latex]{e}^{r+10}-10=-42[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135173523\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135173523\"]\r\n

No solution<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [latex]2\\cdot {10}^{9a}=29[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [latex]-8\\cdot {10}^{p+7}-7=-24[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135188812\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135188812\"]\r\n

[latex]p=\\mathrm{log}\\left(\\frac{17}{8}\\right)-7[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. [latex]7{e}^{3n-5}+5=-89[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17. [latex]{e}^{-3k}+6=44[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137667780\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137667780\"]\r\n

[latex]k=-\\frac{\\mathrm{ln}\\left(38\\right)}{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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18. [latex]-5{e}^{9x-8}-8=-62[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19. [latex]-6{e}^{9x+8}+2=-74[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135253221\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135253221\"]\r\n

[latex]x=\\frac{\\mathrm{ln}\\left(\\frac{38}{3}\\right)-8}{9}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20. [latex]{2}^{x+1}={5}^{2x-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [latex]{e}^{2x}-{e}^{x}-132=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137658150\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137658150\"]\r\n

[latex]x=\\mathrm{ln}12\\text{ }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22. [latex]7{e}^{8x+8}-5=-95[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [latex]10{e}^{8x+3}+2=8[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135475913\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135475913\"]\r\n

[latex]x=\\frac{\\mathrm{ln}\\left(\\frac{3}{5}\\right)-3}{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24. [latex]4{e}^{3x+3}-7=53[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. [latex]8{e}^{-5x-2}-4=-90[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n[reveal-answer q=\"415906\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"415906\"]no solution[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n
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26. [latex]{3}^{2x+1}={7}^{x-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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27. [latex]{e}^{2x}-{e}^{x}-6=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135195715\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135195715\"]\r\n

[latex]x=\\mathrm{ln}\\left(3\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n28. [latex]3{e}^{3-3x}+6=-31[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.<\/p>\r\n\r\n

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\r\n\r\n29. [latex]\\mathrm{log}\\left(\\frac{1}{100}\\right)=-2[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135487112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135487112\"]\r\n

[latex]{10}^{-2}=\\frac{1}{100}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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30. [latex]{\\mathrm{log}}_{324}\\left(18\\right)=\\frac{1}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the definition of a logarithm to solve the equation.<\/p>\r\n\r\n

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31. [latex]5{\\mathrm{log}}_{7}n=10[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137652929\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137652929\"]\r\n

[latex]n=49[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32. [latex]-8{\\mathrm{log}}_{9}x=16[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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33. [latex]4+{\\mathrm{log}}_{2}\\left(9k\\right)=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135570427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135570427\"]\r\n

[latex]k=\\frac{1}{36}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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34. [latex]2\\mathrm{log}\\left(8n+4\\right)+6=10[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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35. [latex]10-4\\mathrm{ln}\\left(9-8x\\right)=6[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134232199\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134232199\"]\r\n

[latex]x=\\frac{9-e}{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises, use the one-to-one property of logarithms to solve.\r\n

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36. [latex]\\mathrm{ln}\\left(10-3x\\right)=\\mathrm{ln}\\left(-4x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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37. [latex]{\\mathrm{log}}_{13}\\left(5n-2\\right)={\\mathrm{log}}_{13}\\left(8-5n\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137424487\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137424487\"]\r\n

[latex]n=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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38. [latex]\\mathrm{log}\\left(x+3\\right)-\\mathrm{log}\\left(x\\right)=\\mathrm{log}\\left(74\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165137436082\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137436082\"]\r\n

No solution<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. [latex]{\\mathrm{log}}_{4}\\left(6-m\\right)={\\mathrm{log}}_{4}3m[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165135511361\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135511361\"]\r\n

No solution<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. [latex]{\\mathrm{log}}_{9}\\left(2{n}^{2}-14n\\right)={\\mathrm{log}}_{9}\\left(-45+{n}^{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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43. [latex]\\mathrm{ln}\\left({x}^{2}-10\\right)+\\mathrm{ln}\\left(9\\right)=\\mathrm{ln}\\left(10\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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44. [latex]x=\u00b1\\frac{10}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, solve each equation for[latex]\\text{ }x.[\/latex]<\/p>\r\n\r\n

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45. [latex]\\mathrm{log}\\left(x+12\\right)=\\mathrm{log}\\left(x\\right)+\\mathrm{log}\\left(12\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165135182927\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135182927\"]\r\n

[latex]x=10[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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47. [latex]{\\mathrm{log}}_{2}\\left(7x+6\\right)=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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48. [latex]\\mathrm{ln}\\left(7\\right)+\\mathrm{ln}\\left(2-4{x}^{2}\\right)=\\mathrm{ln}\\left(14\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n49. [latex]x=0[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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50. [latex]{\\mathrm{log}}_{8}\\left(x+6\\right)-{\\mathrm{log}}_{8}\\left(x\\right)={\\mathrm{log}}_{8}\\left(58\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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51. [latex]\\mathrm{ln}\\left(3\\right)-\\mathrm{ln}\\left(3-3x\\right)=\\mathrm{ln}\\left(4\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137793981\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137793981\"]\r\n

[latex]x=\\frac{3}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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52. [latex]{\\mathrm{log}}_{3}\\left(3x\\right)-{\\mathrm{log}}_{3}\\left(6\\right)={\\mathrm{log}}_{3}\\left(77\\right)[\/latex]<\/p>\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, solve the equation for[latex]\\text{ }x,[\/latex]if there is a solution.<\/em> Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.<\/p>\r\n\r\n

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53. [latex]{\\mathrm{log}}_{9}\\left(x\\right)-5=-4[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135209554\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135209554\"]\r\n

[latex]x=9[\/latex]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n54. [latex]{\\mathrm{log}}_{3}\\left(x\\right)+3=2[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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55. [latex]\\mathrm{ln}\\left(3x\\right)=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137400042\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137400042\"][latex]x=\\frac{{e}^{2}}{3}\\approx 2.5[\/latex]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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56. [latex]\\mathrm{ln}\\left(x-5\\right)=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57. [latex]\\mathrm{log}\\left(4\\right)+\\mathrm{log}\\left(-5x\\right)=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"158343\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"158343\"][latex]x=-5[\/latex]<\/p>\r\n\"Graph[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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58. [latex]-7+{\\mathrm{log}}_{3}\\left(4-x\\right)=-6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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59. [latex]\\mathrm{ln}\\left(4x-10\\right)-6=-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137862576\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137862576\"][latex]x=\\frac{e+10}{4}\\approx 3.2[\/latex]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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60. [latex]\\mathrm{log}\\left(4-2x\\right)=\\mathrm{log}\\left(-4x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n61. [latex]{\\mathrm{log}}_{11}\\left(-2{x}^{2}-7x\\right)={\\mathrm{log}}_{11}\\left(x-2\\right)[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165137834542\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137834542\"]No solution\"Graph[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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62. [latex]\\mathrm{ln}\\left(2x+9\\right)=\\mathrm{ln}\\left(-5x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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63. [latex]{\\mathrm{log}}_{9}\\left(3-x\\right)={\\mathrm{log}}_{9}\\left(4x-8\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135350621\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135350621\"]\r\n

[latex]x=\\frac{11}{5}\\approx 2.2[\/latex]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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64. [latex]\\mathrm{log}\\left({x}^{2}+13\\right)=\\mathrm{log}\\left(7x+3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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65. [latex]\\frac{3}{{\\mathrm{log}}_{2}\\left(10\\right)}-\\mathrm{log}\\left(x-9\\right)=\\mathrm{log}\\left(44\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137526306\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137526306\"]\r\n

[latex]x=\\frac{101}{11}\\approx 9.2[\/latex]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.<\/p>\r\n\r\n

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67. An account with an initial deposit of[latex]\\text{ }\\text{$6,500}\\text{ }[\/latex]earns[latex]\\text{ }7.25%\\text{ }[\/latex]annual interest, compounded continuously. How much will the account be worth after 20 years?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137428225\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137428225\"]\r\n

about[latex]\\text{ }$27,710.24[\/latex]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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68. The formula for measuring sound intensity in decibels[latex]\\text{ }D\\text{ }[\/latex]is defined by the equation[latex]\\text{ }D=10\\mathrm{log}\\left(\\frac{I}{{I}_{0}}\\right),\\text{}[\/latex]where[latex]\\text{ }I\\text{ }[\/latex]is the intensity of the sound in watts per square meter and[latex]\\text{ }{I}_{0}={10}^{-12}\\text{ }[\/latex]is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of[latex]\\text{ }8.3\\cdot {10}^{2}\\text{ }[\/latex]watts per square meter?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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69. The population of a small town is modeled by the equation[latex]\\text{ }P=1650{e}^{0.5t}\\text{ }[\/latex]where[latex]\\text{ }t\\text{ }[\/latex]is measured in years. In approximately how many years will the town\u2019s population reach[latex]\\text{ }\\text{20,000?}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137737687\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137737687\"]\r\n

about 5 years<\/p>\r\n\"Graph[\/hidden-answer]\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, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.<\/p>\r\n\r\n

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\r\n\r\n70. [latex]1000{\\left(1.03\\right)}^{t}=5000\\text{ }[\/latex]using the common log.\r\n\r\n<\/div>\r\n<\/div>\r\n
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71. [latex]{e}^{5x}=17\\text{ }[\/latex]using the natural log<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n72. [latex]\\frac{\\mathrm{ln}\\left(17\\right)}{5}\\approx 0.567[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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73. [latex]3{\\left(1.04\\right)}^{3t}=8\\text{ }[\/latex]using the common log<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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74. [latex]{3}^{4x-5}=38\\text{ }[\/latex]using the common log<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134267805\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134267805\"]\r\n

[latex]x=\\frac{\\mathrm{log}\\left(38\\right)+5\\mathrm{log}\\left(3\\right)\\text{ }}{4\\mathrm{log}\\left(3\\right)}\\approx 2.078[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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75. [latex]50{e}^{-0.12t}=10\\text{ }[\/latex]using the natural log<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.<\/p>\r\n\r\n

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76. [latex]7{e}^{3x-5}+7.9=47[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137653974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137653974\"]\r\n

[latex]x\\approx 2.2401[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n77. [latex]\\mathrm{ln}\\left(3\\right)+\\mathrm{ln}\\left(4.4x+6.8\\right)=2[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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78. [latex]\\mathrm{log}\\left(-0.7x-9\\right)=1+5\\mathrm{log}\\left(5\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134319718\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134319718\"]\r\n

[latex]x\\approx -\\text{44655}.\\text{7143}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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79. Atmospheric pressure[latex]\\text{ }P\\text{ }[\/latex]in pounds per square inch is represented by the formula[latex]\\text{ }P=14.7{e}^{-0.21x},[\/latex] where [latex]x[\/latex] is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of[latex]\\text{ }8.369\\text{ }[\/latex]pounds per square inch? (Hint<\/em>: there are 5280 feet in a mile)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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80. The magnitude M <\/em>of an earthquake is represented by the equation[latex]\\text{ }M=\\frac{2}{3}\\mathrm{log}\\left(\\frac{E}{{E}_{0}}\\right)\\text{ }[\/latex]where[latex]\\text{ }E\\text{ }[\/latex]is the amount of energy released by the earthquake in joules and[latex]\\text{ }{E}_{0}={10}^{4.4}\\text{ }[\/latex]is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing[latex]\\text{ }1.4\\cdot {10}^{13}\\text{ }[\/latex]joules of energy?<\/p>\r\n\r\n<\/div>\r\n

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about[latex]\\text{ }5.83[\/latex]<\/p>\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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81. Use the definition of a logarithm along with the one-to-one property of logarithms to prove that [latex]\\text{ }{b}^{{\\mathrm{log}}_{b}x}=x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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82. Recall the formula for continually compounding interest,[latex]\\text{ }y=A{e}^{kt}.\\text{ }[\/latex]Use the definition of a logarithm along with properties of logarithms to solve the formula for time[latex]\\text{ }t\\text{ }[\/latex]such that[latex]\\text{ }t\\text{ }[\/latex]is equal to a single logarithm.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135191295\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135191295\"]\r\n

[latex]t=\\mathrm{ln}\\left({\\left(\\frac{y}{A}\\right)}^{\\frac{1}{k}}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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83. Recall the compound interest formula[latex]\\text{ }A=a{\\left(1+\\frac{r}{k}\\right)}^{kt}.\\text{ }[\/latex]Use the definition of a logarithm along with properties of logarithms to solve the formula for time[latex]\\text{ }t.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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84. Newton\u2019s Law of Cooling states that the temperature[latex]\\text{ }T\\text{ }[\/latex]of an object at any time t<\/em> can be described by the equation[latex]\\text{ }T={T}_{s}+\\left({T}_{0}-{T}_{s}\\right){e}^{-kt},[\/latex] where[latex]\\text{ }{T}_{s}\\text{ }[\/latex]is the temperature of the surrounding environment,[latex]\\text{ }{T}_{0}\\text{ }[\/latex]is the initial temperature of the object, and[latex]\\text{ }k\\text{}[\/latex]is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time[latex]\\text{ }t\\text{ }[\/latex]such that[latex]\\text{ }t\\text{ }[\/latex]is equal to a single logarithm.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137426905\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137426905\"]\r\n

[latex]t=\\mathrm{ln}\\left({\\left(\\frac{T-{T}_{s}}{{T}_{0}-{T}_{s}}\\right)}^{-\\text{ }\\frac{1}{k}}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

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2.5 Section Exercises<\/h3>\n
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Verbal<\/h4>\n
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1. How can an exponential equation be solved?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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2. When does an extraneous solution occur? How can an extraneous solution be recognized?<\/p>\n<\/div>\n<\/div>\n

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3. When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Algebraic<\/h4>\n

For the following exercises, use like bases to solve the exponential equation.<\/p>\n

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4. [latex]{4}^{-3v-2}={4}^{-v}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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5. [latex]64\\cdot {4}^{3x}=16[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=-\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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6. [latex]{3}^{2x+1}\\cdot {3}^{x}=243[\/latex]<\/p>\n<\/div>\n<\/div>\n

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7. [latex]{2}^{-3n}\\cdot \\frac{1}{4}={2}^{n+2}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]n=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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8. [latex]625\\cdot {5}^{3x+3}=125[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9. [latex]\\frac{{36}^{3b}}{{36}^{2b}}={216}^{2-b}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]b=\\frac{6}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10. [latex]{\\left(\\frac{1}{64}\\right)}^{3n}\\cdot 8={2}^{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, use logarithms to solve.<\/p>\n

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11. [latex]{9}^{x-10}=1[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12. [latex]2{e}^{6x}=13[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13. [latex]{e}^{r+10}-10=-42[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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No solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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14. [latex]2\\cdot {10}^{9a}=29[\/latex]<\/p>\n<\/div>\n<\/div>\n

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15. [latex]-8\\cdot {10}^{p+7}-7=-24[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]p=\\mathrm{log}\\left(\\frac{17}{8}\\right)-7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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16. [latex]7{e}^{3n-5}+5=-89[\/latex]<\/p>\n<\/div>\n<\/div>\n

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17. [latex]{e}^{-3k}+6=44[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]k=-\\frac{\\mathrm{ln}\\left(38\\right)}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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18. [latex]-5{e}^{9x-8}-8=-62[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19. [latex]-6{e}^{9x+8}+2=-74[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=\\frac{\\mathrm{ln}\\left(\\frac{38}{3}\\right)-8}{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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20. [latex]{2}^{x+1}={5}^{2x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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21. [latex]{e}^{2x}-{e}^{x}-132=0[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=\\mathrm{ln}12\\text{ }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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22. [latex]7{e}^{8x+8}-5=-95[\/latex]<\/p>\n<\/div>\n<\/div>\n

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23. [latex]10{e}^{8x+3}+2=8[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=\\frac{\\mathrm{ln}\\left(\\frac{3}{5}\\right)-3}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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24. [latex]4{e}^{3x+3}-7=53[\/latex]<\/p>\n<\/div>\n<\/div>\n

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25. [latex]8{e}^{-5x-2}-4=-90[\/latex]<\/p>\n<\/div>\n

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Show Answer<\/span><\/p>\n
no solution<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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26. [latex]{3}^{2x+1}={7}^{x-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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27. [latex]{e}^{2x}-{e}^{x}-6=0[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=\\mathrm{ln}\\left(3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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28. [latex]3{e}^{3-3x}+6=-31[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.<\/p>\n

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29. [latex]\\mathrm{log}\\left(\\frac{1}{100}\\right)=-2[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]{10}^{-2}=\\frac{1}{100}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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30. [latex]{\\mathrm{log}}_{324}\\left(18\\right)=\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, use the definition of a logarithm to solve the equation.<\/p>\n

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31. [latex]5{\\mathrm{log}}_{7}n=10[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]n=49[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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32. [latex]-8{\\mathrm{log}}_{9}x=16[\/latex]<\/p>\n<\/div>\n<\/div>\n

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33. [latex]4+{\\mathrm{log}}_{2}\\left(9k\\right)=2[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]k=\\frac{1}{36}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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34. [latex]2\\mathrm{log}\\left(8n+4\\right)+6=10[\/latex]<\/p>\n<\/div>\n<\/div>\n

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35. [latex]10-4\\mathrm{ln}\\left(9-8x\\right)=6[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=\\frac{9-e}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, use the one-to-one property of logarithms to solve.<\/p>\n

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36. [latex]\\mathrm{ln}\\left(10-3x\\right)=\\mathrm{ln}\\left(-4x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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37. [latex]{\\mathrm{log}}_{13}\\left(5n-2\\right)={\\mathrm{log}}_{13}\\left(8-5n\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]n=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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38. [latex]\\mathrm{log}\\left(x+3\\right)-\\mathrm{log}\\left(x\\right)=\\mathrm{log}\\left(74\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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No solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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40. [latex]{\\mathrm{log}}_{4}\\left(6-m\\right)={\\mathrm{log}}_{4}3m[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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No solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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42. [latex]{\\mathrm{log}}_{9}\\left(2{n}^{2}-14n\\right)={\\mathrm{log}}_{9}\\left(-45+{n}^{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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43. [latex]\\mathrm{ln}\\left({x}^{2}-10\\right)+\\mathrm{ln}\\left(9\\right)=\\mathrm{ln}\\left(10\\right)[\/latex]<\/p>\n<\/div>\n

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44. [latex]x=\u00b1\\frac{10}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, solve each equation for[latex]\\text{ }x.[\/latex]<\/p>\n

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45. [latex]\\mathrm{log}\\left(x+12\\right)=\\mathrm{log}\\left(x\\right)+\\mathrm{log}\\left(12\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
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[latex]x=10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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47. [latex]{\\mathrm{log}}_{2}\\left(7x+6\\right)=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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48. [latex]\\mathrm{ln}\\left(7\\right)+\\mathrm{ln}\\left(2-4{x}^{2}\\right)=\\mathrm{ln}\\left(14\\right)[\/latex]<\/p>\n<\/div>\n

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49. [latex]x=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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50. [latex]{\\mathrm{log}}_{8}\\left(x+6\\right)-{\\mathrm{log}}_{8}\\left(x\\right)={\\mathrm{log}}_{8}\\left(58\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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51. [latex]\\mathrm{ln}\\left(3\\right)-\\mathrm{ln}\\left(3-3x\\right)=\\mathrm{ln}\\left(4\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=\\frac{3}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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52. [latex]{\\mathrm{log}}_{3}\\left(3x\\right)-{\\mathrm{log}}_{3}\\left(6\\right)={\\mathrm{log}}_{3}\\left(77\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Graphical<\/h4>\n

For the following exercises, solve the equation for[latex]\\text{ }x,[\/latex]if there is a solution.<\/em> Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.<\/p>\n

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53. [latex]{\\mathrm{log}}_{9}\\left(x\\right)-5=-4[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=9[\/latex]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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Show Solution<\/span><\/p>\n
[latex]x=\\frac{{e}^{2}}{3}\\approx 2.5[\/latex]\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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56. [latex]\\mathrm{ln}\\left(x-5\\right)=1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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57. [latex]\\mathrm{log}\\left(4\\right)+\\mathrm{log}\\left(-5x\\right)=2[\/latex]<\/p>\n<\/div>\n

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Show Answer<\/span><\/p>\n
[latex]x=-5[\/latex]<\/p>\n

\"Graph<\/div>\n<\/div>\n

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

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58. [latex]-7+{\\mathrm{log}}_{3}\\left(4-x\\right)=-6[\/latex]<\/p>\n<\/div>\n<\/div>\n

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59. [latex]\\mathrm{ln}\\left(4x-10\\right)-6=-5[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]x=\\frac{e+10}{4}\\approx 3.2[\/latex]\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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60. [latex]\\mathrm{log}\\left(4-2x\\right)=\\mathrm{log}\\left(-4x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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61. [latex]{\\mathrm{log}}_{11}\\left(-2{x}^{2}-7x\\right)={\\mathrm{log}}_{11}\\left(x-2\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
No solution\"Graph<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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62. [latex]\\mathrm{ln}\\left(2x+9\\right)=\\mathrm{ln}\\left(-5x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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63. [latex]{\\mathrm{log}}_{9}\\left(3-x\\right)={\\mathrm{log}}_{9}\\left(4x-8\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=\\frac{11}{5}\\approx 2.2[\/latex]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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64. [latex]\\mathrm{log}\\left({x}^{2}+13\\right)=\\mathrm{log}\\left(7x+3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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65. [latex]\\frac{3}{{\\mathrm{log}}_{2}\\left(10\\right)}-\\mathrm{log}\\left(x-9\\right)=\\mathrm{log}\\left(44\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=\\frac{101}{11}\\approx 9.2[\/latex]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

For the following exercises, solve for the indicated value, and graph the situation showing the solution point.<\/p>\n

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67. An account with an initial deposit of[latex]\\text{ }\\text{$6,500}\\text{ }[\/latex]earns[latex]\\text{ }7.25%\\text{ }[\/latex]annual interest, compounded continuously. How much will the account be worth after 20 years?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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about[latex]\\text{ }$27,710.24[\/latex]<\/p>\n

\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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68. The formula for measuring sound intensity in decibels[latex]\\text{ }D\\text{ }[\/latex]is defined by the equation[latex]\\text{ }D=10\\mathrm{log}\\left(\\frac{I}{{I}_{0}}\\right),\\text{}[\/latex]where[latex]\\text{ }I\\text{ }[\/latex]is the intensity of the sound in watts per square meter and[latex]\\text{ }{I}_{0}={10}^{-12}\\text{ }[\/latex]is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of[latex]\\text{ }8.3\\cdot {10}^{2}\\text{ }[\/latex]watts per square meter?<\/p>\n<\/div>\n<\/div>\n

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69. The population of a small town is modeled by the equation[latex]\\text{ }P=1650{e}^{0.5t}\\text{ }[\/latex]where[latex]\\text{ }t\\text{ }[\/latex]is measured in years. In approximately how many years will the town\u2019s population reach[latex]\\text{ }\\text{20,000?}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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about 5 years<\/p>\n

\"Graph<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Technology<\/h4>\n

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.<\/p>\n

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70. [latex]1000{\\left(1.03\\right)}^{t}=5000\\text{ }[\/latex]using the common log.<\/p>\n<\/div>\n<\/div>\n

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71. [latex]{e}^{5x}=17\\text{ }[\/latex]using the natural log<\/p>\n<\/div>\n

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72. [latex]\\frac{\\mathrm{ln}\\left(17\\right)}{5}\\approx 0.567[\/latex]<\/p>\n<\/div>\n<\/div>\n

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73. [latex]3{\\left(1.04\\right)}^{3t}=8\\text{ }[\/latex]using the common log<\/p>\n<\/div>\n<\/div>\n

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74. [latex]{3}^{4x-5}=38\\text{ }[\/latex]using the common log<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x=\\frac{\\mathrm{log}\\left(38\\right)+5\\mathrm{log}\\left(3\\right)\\text{ }}{4\\mathrm{log}\\left(3\\right)}\\approx 2.078[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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75. [latex]50{e}^{-0.12t}=10\\text{ }[\/latex]using the natural log<\/p>\n<\/div>\n<\/div>\n

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.<\/p>\n

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76. [latex]7{e}^{3x-5}+7.9=47[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x\\approx 2.2401[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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77. [latex]\\mathrm{ln}\\left(3\\right)+\\mathrm{ln}\\left(4.4x+6.8\\right)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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78. [latex]\\mathrm{log}\\left(-0.7x-9\\right)=1+5\\mathrm{log}\\left(5\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]x\\approx -\\text{44655}.\\text{7143}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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79. Atmospheric pressure[latex]\\text{ }P\\text{ }[\/latex]in pounds per square inch is represented by the formula[latex]\\text{ }P=14.7{e}^{-0.21x},[\/latex] where [latex]x[\/latex] is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of[latex]\\text{ }8.369\\text{ }[\/latex]pounds per square inch? (Hint<\/em>: there are 5280 feet in a mile)<\/p>\n<\/div>\n<\/div>\n

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80. The magnitude M <\/em>of an earthquake is represented by the equation[latex]\\text{ }M=\\frac{2}{3}\\mathrm{log}\\left(\\frac{E}{{E}_{0}}\\right)\\text{ }[\/latex]where[latex]\\text{ }E\\text{ }[\/latex]is the amount of energy released by the earthquake in joules and[latex]\\text{ }{E}_{0}={10}^{4.4}\\text{ }[\/latex]is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing[latex]\\text{ }1.4\\cdot {10}^{13}\\text{ }[\/latex]joules of energy?<\/p>\n<\/div>\n

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about[latex]\\text{ }5.83[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Extensions<\/h4>\n
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81. Use the definition of a logarithm along with the one-to-one property of logarithms to prove that [latex]\\text{ }{b}^{{\\mathrm{log}}_{b}x}=x.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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82. Recall the formula for continually compounding interest,[latex]\\text{ }y=A{e}^{kt}.\\text{ }[\/latex]Use the definition of a logarithm along with properties of logarithms to solve the formula for time[latex]\\text{ }t\\text{ }[\/latex]such that[latex]\\text{ }t\\text{ }[\/latex]is equal to a single logarithm.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]t=\\mathrm{ln}\\left({\\left(\\frac{y}{A}\\right)}^{\\frac{1}{k}}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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83. Recall the compound interest formula[latex]\\text{ }A=a{\\left(1+\\frac{r}{k}\\right)}^{kt}.\\text{ }[\/latex]Use the definition of a logarithm along with properties of logarithms to solve the formula for time[latex]\\text{ }t.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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84. Newton\u2019s Law of Cooling states that the temperature[latex]\\text{ }T\\text{ }[\/latex]of an object at any time t<\/em> can be described by the equation[latex]\\text{ }T={T}_{s}+\\left({T}_{0}-{T}_{s}\\right){e}^{-kt},[\/latex] where[latex]\\text{ }{T}_{s}\\text{ }[\/latex]is the temperature of the surrounding environment,[latex]\\text{ }{T}_{0}\\text{ }[\/latex]is the initial temperature of the object, and[latex]\\text{ }k\\text{}[\/latex]is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time[latex]\\text{ }t\\text{ }[\/latex]such that[latex]\\text{ }t\\text{ }[\/latex]is equal to a single logarithm.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]t=\\mathrm{ln}\\left({\\left(\\frac{T-{T}_{s}}{{T}_{0}-{T}_{s}}\\right)}^{-\\text{ }\\frac{1}{k}}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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