{"id":2366,"date":"2019-05-13T13:05:04","date_gmt":"2019-05-13T13:05:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2366"},"modified":"2019-05-13T13:05:15","modified_gmt":"2019-05-13T13:05:15","slug":"2-4-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/2-4-section-exercises\/","title":{"raw":"2.4 Section Exercises","rendered":"2.4 Section Exercises"},"content":{"raw":"
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2.4 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. What is a base[latex]\\text{ }b\\text{ }[\/latex]logarithm? Discuss the meaning by interpreting each part of the equivalent equations[latex]\\text{ }{b}^{y}=x\\text{ }[\/latex]and[latex]\\text{ }{\\mathrm{log}}_{b}x=y\\text{ }[\/latex]for[latex]\\text{ }b>0,b\\ne 1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137411524\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137411524\"]A logarithm is an exponent. Specifically, it is the exponent to which a base[latex]\\text{ }b\\text{ }[\/latex]is raised to produce a given value. In the expressions given, the base[latex]\\text{ }b\\text{ }[\/latex]has the same value. The exponent,[latex]\\text{ }y,[\/latex]in the expression[latex]\\text{ }{b}^{y}\\text{ }[\/latex]can also be written as the logarithm,[latex]\\text{ }{\\mathrm{log}}_{b}x,[\/latex]and the value of[latex]\\text{ }x\\text{ }[\/latex]is the result of raising[latex]\\text{ }b\\text{ }[\/latex]to the power of[latex]\\text{ }y.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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2. How is the logarithmic function[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}x\\text{ }[\/latex]related to the exponential function[latex]\\text{ }g\\left(x\\right)={b}^{x}?\\text{ }[\/latex]What is the result of composing these two functions?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. How can the logarithmic equation[latex]\\text{ }{\\mathrm{log}}_{b}x=y\\text{ }[\/latex]be solved for[latex]\\text{ }x\\text{ }[\/latex]using the properties of exponents?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135203815\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135203815\"]\r\n

Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation[latex]\\text{ }{b}^{y}=x,[\/latex] and then properties of exponents can be applied to solve for[latex]\\text{ }x.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base[latex]\\text{ }b,[\/latex] and how does the notation differ?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5. Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base[latex]\\text{ }b,[\/latex] and how does the notation differ?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"341017\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"341017\"]The natural logarithm is a special case of the logarithm with base[latex]\\text{ }b\\text{ }[\/latex]in that the natural log always has base[latex]\\text{ }e.\\text{ }[\/latex]Rather than notating the natural logarithm as[latex]\\text{ }{\\mathrm{log}}_{e}\\left(x\\right),[\/latex]the notation used is[latex]\\text{ }\\mathrm{ln}\\left(x\\right).[\/latex][\/hidden-answer]<\/p>\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, rewrite each equation in exponential form.<\/p>\r\n\r\n

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

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7. [latex]{\\text{log}}_{a}\\left(b\\right)=c[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135191877\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135191877\"]\r\n

[latex]{a}^{c}=b[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8. [latex]{\\mathrm{log}}_{16}\\left(y\\right)=x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [latex]{\\mathrm{log}}_{x}\\left(64\\right)=y[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137428120\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137428120\"]\r\n

[latex]{x}^{y}=64[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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11. [latex]{\\mathrm{log}}_{15}\\left(a\\right)=b[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135191871\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135191871\"][latex]{15}^{b}=a[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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12. [latex]{\\mathrm{log}}_{y}\\left(137\\right)=x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [latex]{\\mathrm{log}}_{13}\\left(142\\right)=a[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137580134\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137580134\"]\r\n

[latex]{13}^{a}=142[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [latex]\\text{log}\\left(v\\right)=t[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [latex]\\text{ln}\\left(w\\right)=n[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135185210\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135185210\"]\r\n

[latex]{e}^{n}=w[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, rewrite each equation in logarithmic form.<\/p>\r\n\r\n

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16. [latex]{4}^{x}=y[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17. [latex]{c}^{d}=k[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137742265\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137742265\"]\r\n

[latex]{\\text{log}}_{c}\\left(k\\right)=d[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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18. [latex]{m}^{-7}=n[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19. [latex]{19}^{x}=y[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135296129\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135296129\"]\r\n

[latex]{\\mathrm{log}}_{19}y=x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20. [latex]{x}^{-\\text{ }\\frac{10}{13}}=y[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [latex]{n}^{4}=103[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134032258\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134032258\"]\r\n

[latex]{\\mathrm{log}}_{n}\\left(103\\right)=4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22. [latex]{\\left(\\frac{7}{5}\\right)}^{m}=n[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [latex]{y}^{x}=\\frac{39}{100}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135194466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135194466\"]\r\n

[latex]{\\mathrm{log}}_{y}\\left(\\frac{39}{100}\\right)=x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24. [latex]{10}^{a}=b[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. [latex]{e}^{k}=h[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135191801\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135191801\"]\r\n

[latex]\\text{ln}\\left(h\\right)=k[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, solve for[latex]\\text{ }x\\text{ }[\/latex]by converting the logarithmic equation to exponential form.<\/p>\r\n\r\n

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26. [latex]{\\text{log}}_{3}\\left(x\\right)=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165137443266\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137443266\"]\r\n

[latex]x={2}^{-3}=\\frac{1}{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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28. [latex]{\\text{log}}_{5}\\left(x\\right)=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165137409760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137409760\"]\r\n

[latex]x={3}^{3}=27[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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30. [latex]{\\text{log}}_{2}\\left(x\\right)=6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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31. [latex]{\\text{log}}_{9}\\left(x\\right)=\\frac{1}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137469004\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137469004\"]\r\n

[latex]x={9}^{\\frac{1}{2}}=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32. [latex]{\\text{log}}_{18}\\left(x\\right)=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165135253754\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135253754\"]\r\n

[latex]x={6}^{-3}=\\frac{1}{216}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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34. [latex]\\text{log}\\left(x\\right)=3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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35. [latex]\\text{ln}\\left(x\\right)=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135524549\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135524549\"]\r\n

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

For the following exercises, use the definition of common and natural logarithms to simplify.<\/p>\r\n\r\n

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36. [latex]\\text{log}\\left({100}^{8}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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37. [latex]{10}^{\\text{log}\\left(32\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137436613\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137436613\"]\r\n

[latex]32[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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38. [latex]2\\text{log}\\left(.0001\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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39. [latex]{e}^{\\mathrm{ln}\\left(1.06\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137455559\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137455559\"]\r\n

[latex]1.06[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. [latex]\\mathrm{ln}\\left({e}^{-5.03}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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41. [latex]{e}^{\\mathrm{ln}\\left(10.125\\right)}+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137605063\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137605063\"][latex]14.125[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.<\/p>\r\n\r\n

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

[reveal-answer q=\"fs-id1165137843718\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137843718\"]\r\n

[latex]{\\mathrm{log}}_{b}\\left(2\\right)+{\\mathrm{log}}_{b}\\left(7\\right)+{\\mathrm{log}}_{b}\\left(x\\right)+{\\mathrm{log}}_{b}\\left(y\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165135356586\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135356586\"]\r\n

[latex]{\\mathrm{log}}_{b}\\left(13\\right)-{\\mathrm{log}}_{b}\\left(17\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165134550670\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134550670\"]\r\n

[latex]-k\\mathrm{ln}\\left(4\\right)[\/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({y}^{x}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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

[reveal-answer q=\"fs-id1165134055933\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134055933\"]\r\n

[latex]\\mathrm{ln}\\left(7xy\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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49. [latex]{\\mathrm{log}}_{3}\\left(2\\right)+{\\mathrm{log}}_{3}\\left(a\\right)+{\\mathrm{log}}_{3}\\left(11\\right)+{\\mathrm{log}}_{3}\\left(b\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165135255369\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135255369\"]\r\n

[latex]{\\mathrm{log}}_{b}\\left(4\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165137834064\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137834064\"]\r\n

[latex]{\\text{log}}_{b}\\left(7\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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Numeric<\/h4>\r\n

For the following exercises, evaluate the base[latex]\\text{ }b\\text{ }[\/latex]logarithmic expression without using a calculator.<\/p>\r\n\r\n

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54. [latex]{\\text{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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55. [latex]{\\text{log}}_{6}\\left(\\sqrt{6}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135706822\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135706822\"]\r\n

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

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56. [latex]{\\text{log}}_{2}\\left(\\frac{1}{8}\\right)+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57. [latex]6{\\text{log}}_{8}\\left(4\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137454533\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137454533\"]\r\n

[latex]4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, evaluate the common logarithmic expression without using a calculator.<\/p>\r\n\r\n

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58. [latex]\\text{log}\\left(10,000\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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59. [latex]\\text{log}\\left(0.001\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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60. [latex]-\\text{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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61. [latex]\\text{log}\\left(1\\right)+7[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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62. [latex]2\\text{log}\\left({100}^{-3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137588216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137588216\"]\r\n

[latex]-12[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, evaluate the natural logarithmic expression without using a calculator.<\/p>\r\n\r\n

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63. [latex]\\text{ln}\\left({e}^{\\frac{1}{3}}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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64. [latex]\\text{ln}\\left(1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137645466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137645466\"]\r\n

[latex]0[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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65. [latex]\\text{ln}\\left({e}^{-0.225}\\right)-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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66. [latex]25\\text{ln}\\left({e}^{\\frac{2}{5}}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137418633\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137418633\"]\r\n

[latex]10[\/latex]<\/p>\r\n[\/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, evaluate each expression using a calculator. Round to the nearest thousandth.<\/p>\r\n\r\n

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67. [latex]\\text{log}\\left(0.04\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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68. [latex]\\text{ln}\\left(15\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137696603\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137696603\"]\r\n

[latex]\\text{2}.\\text{7}0\\text{8}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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69. [latex]\\text{ln}\\left(\\frac{4}{5}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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70. [latex]\\text{log}\\left(\\sqrt{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137658294\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137658294\"]\r\n

[latex]0.151[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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71. [latex]\\text{ln}\\left(\\sqrt{2}\\right)[\/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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72. Is[latex]\\text{ }x=0\\text{ }[\/latex]in the domain of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(x\\right)?\\text{ }[\/latex]If so, what is the value of the function when[latex]\\text{ }x=0?\\text{ }[\/latex]Verify the result.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137474123\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137474123\"]\r\n

No, the function has no defined value for[latex]\\text{ }x=0.\\text{ }[\/latex]To verify, suppose[latex]\\text{ }x=0\\text{ }[\/latex]is in the domain of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(x\\right).\\text{ }[\/latex]Then there is some number[latex]\\text{ }n\\text{ }[\/latex]such that[latex]\\text{ }n=\\mathrm{log}\\left(0\\right).\\text{ }[\/latex]Rewriting as an exponential equation gives:[latex]\\text{ }{10}^{n}=0,[\/latex] which is impossible since no such real number[latex]\\text{ }n\\text{ }[\/latex]exists. Therefore,[latex]\\text{ }x=0\\text{ }[\/latex]is not<\/em> the domain of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(x\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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73. Is[latex]\\text{ }f\\left(x\\right)=0\\text{ }[\/latex]in the range of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(x\\right)?\\text{ }[\/latex]If so, for what value of[latex]\\text{ }x?\\text{ }[\/latex]Verify the result.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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74. Is there a number[latex]\\text{ }x\\text{ }[\/latex]such that[latex]\\text{ }\\mathrm{ln}x=2?\\text{ }[\/latex]If so, what is that number? Verify the result.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137659786\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137659786\"]\r\n

Yes. Suppose there exists a real number[latex]\\text{ }x\\text{ }[\/latex]such that[latex]\\text{ }\\mathrm{ln}x=2.\\text{ }[\/latex]Rewriting as an exponential equation gives[latex]\\text{ }x={e}^{2},[\/latex] which is a real number. To verify, let[latex]\\text{ }x={e}^{2}.\\text{ }[\/latex]Then, by definition,[latex]\\text{ }\\mathrm{ln}\\left(x\\right)=\\mathrm{ln}\\left({e}^{2}\\right)=2.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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75. Is the following true:[latex]\\text{ }\\frac{{\\mathrm{log}}_{3}\\left(27\\right)}{{\\mathrm{log}}_{4}\\left(\\frac{1}{64}\\right)}=-1?\\text{ }[\/latex]Verify the result.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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76. Is the following true:[latex]\\text{ }\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}=1.725?\\text{ }[\/latex]Verify the result.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137749907\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137749907\"]\r\n

No;[latex]\\text{ }\\mathrm{ln}\\left(1\\right)=0,[\/latex] so[latex]\\text{ }\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}\\text{ }[\/latex]is undefined.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Real-World Applications<\/h4>\r\n
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77. The exposure index[latex]\\text{ }EI\\text{ }[\/latex]for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation[latex]\\text{ }EI={\\mathrm{log}}_{2}\\left(\\frac{{f}^{2}}{t}\\right),[\/latex] where[latex]\\text{ }f\\text{ }[\/latex]is the \u201cf-stop\u201d setting on the camera, and [latex]t[\/latex] is the exposure time in seconds. Suppose the f-stop setting is[latex]\\text{ }8\\text{ }[\/latex]and the desired exposure time is[latex]\\text{ }2\\text{ }[\/latex]seconds. What will the resulting exposure index be?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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78. Refer to the previous exercise. Suppose the light meter on a camera indicates an[latex]\\text{ }EI\\text{ }[\/latex]of[latex]\\text{ }-2,[\/latex] and the desired exposure time is 16 seconds. What should the f-stop setting be?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137570610\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137570610\"]\r\n

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

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79. The intensity levels I<\/em> of two earthquakes measured on a seismograph can be compared by the formula[latex]\\text{ }\\mathrm{log}\\frac{{I}_{1}}{{I}_{2}}={M}_{1}-{M}_{2}\\text{ }[\/latex]where[latex]\\text{ }M\\text{ }[\/latex]is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0[footnote]http:\/\/earthquake.usgs.gov\/earthquakes\/world\/historical.php. Accessed 3\/4\/2014.[\/footnote].\u00a0How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

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2.4 Section Exercises<\/h3>\n
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Verbal<\/h4>\n
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1. What is a base[latex]\\text{ }b\\text{ }[\/latex]logarithm? Discuss the meaning by interpreting each part of the equivalent equations[latex]\\text{ }{b}^{y}=x\\text{ }[\/latex]and[latex]\\text{ }{\\mathrm{log}}_{b}x=y\\text{ }[\/latex]for[latex]\\text{ }b>0,b\\ne 1.[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
A logarithm is an exponent. Specifically, it is the exponent to which a base[latex]\\text{ }b\\text{ }[\/latex]is raised to produce a given value. In the expressions given, the base[latex]\\text{ }b\\text{ }[\/latex]has the same value. The exponent,[latex]\\text{ }y,[\/latex]in the expression[latex]\\text{ }{b}^{y}\\text{ }[\/latex]can also be written as the logarithm,[latex]\\text{ }{\\mathrm{log}}_{b}x,[\/latex]and the value of[latex]\\text{ }x\\text{ }[\/latex]is the result of raising[latex]\\text{ }b\\text{ }[\/latex]to the power of[latex]\\text{ }y.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2. How is the logarithmic function[latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}x\\text{ }[\/latex]related to the exponential function[latex]\\text{ }g\\left(x\\right)={b}^{x}?\\text{ }[\/latex]What is the result of composing these two functions?<\/p>\n<\/div>\n<\/div>\n

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3. How can the logarithmic equation[latex]\\text{ }{\\mathrm{log}}_{b}x=y\\text{ }[\/latex]be solved for[latex]\\text{ }x\\text{ }[\/latex]using the properties of exponents?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation[latex]\\text{ }{b}^{y}=x,[\/latex] and then properties of exponents can be applied to solve for[latex]\\text{ }x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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4. Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base[latex]\\text{ }b,[\/latex] and how does the notation differ?<\/p>\n<\/div>\n<\/div>\n

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5. Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base[latex]\\text{ }b,[\/latex] and how does the notation differ?<\/p>\n<\/div>\n

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Show Answer<\/span><\/p>\n
The natural logarithm is a special case of the logarithm with base[latex]\\text{ }b\\text{ }[\/latex]in that the natural log always has base[latex]\\text{ }e.\\text{ }[\/latex]Rather than notating the natural logarithm as[latex]\\text{ }{\\mathrm{log}}_{e}\\left(x\\right),[\/latex]the notation used is[latex]\\text{ }\\mathrm{ln}\\left(x\\right).[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Algebraic<\/h4>\n

For the following exercises, rewrite each equation in exponential form.<\/p>\n

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6. [latex]{\\text{log}}_{4}\\left(q\\right)=m[\/latex]<\/p>\n<\/div>\n<\/div>\n

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7. [latex]{\\text{log}}_{a}\\left(b\\right)=c[\/latex]<\/p>\n<\/div>\n

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[latex]{a}^{c}=b[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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8. [latex]{\\mathrm{log}}_{16}\\left(y\\right)=x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9. [latex]{\\mathrm{log}}_{x}\\left(64\\right)=y[\/latex]<\/p>\n<\/div>\n

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[latex]{x}^{y}=64[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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11. [latex]{\\mathrm{log}}_{15}\\left(a\\right)=b[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]{15}^{b}=a[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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12. [latex]{\\mathrm{log}}_{y}\\left(137\\right)=x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13. [latex]{\\mathrm{log}}_{13}\\left(142\\right)=a[\/latex]<\/p>\n<\/div>\n

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[latex]{13}^{a}=142[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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14. [latex]\\text{log}\\left(v\\right)=t[\/latex]<\/p>\n<\/div>\n<\/div>\n

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15. [latex]\\text{ln}\\left(w\\right)=n[\/latex]<\/p>\n<\/div>\n

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[latex]{e}^{n}=w[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, rewrite each equation in logarithmic form.<\/p>\n

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16. [latex]{4}^{x}=y[\/latex]<\/p>\n<\/div>\n<\/div>\n

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17. [latex]{c}^{d}=k[\/latex]<\/p>\n<\/div>\n

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[latex]{\\text{log}}_{c}\\left(k\\right)=d[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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18. [latex]{m}^{-7}=n[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19. [latex]{19}^{x}=y[\/latex]<\/p>\n<\/div>\n

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[latex]{\\mathrm{log}}_{19}y=x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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20. [latex]{x}^{-\\text{ }\\frac{10}{13}}=y[\/latex]<\/p>\n<\/div>\n<\/div>\n

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21. [latex]{n}^{4}=103[\/latex]<\/p>\n<\/div>\n

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[latex]{\\mathrm{log}}_{n}\\left(103\\right)=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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22. [latex]{\\left(\\frac{7}{5}\\right)}^{m}=n[\/latex]<\/p>\n<\/div>\n<\/div>\n

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23. [latex]{y}^{x}=\\frac{39}{100}[\/latex]<\/p>\n<\/div>\n

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[latex]{\\mathrm{log}}_{y}\\left(\\frac{39}{100}\\right)=x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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24. [latex]{10}^{a}=b[\/latex]<\/p>\n<\/div>\n<\/div>\n

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25. [latex]{e}^{k}=h[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{ln}\\left(h\\right)=k[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, solve for[latex]\\text{ }x\\text{ }[\/latex]by converting the logarithmic equation to exponential form.<\/p>\n

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

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

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[latex]x={2}^{-3}=\\frac{1}{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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28. [latex]{\\text{log}}_{5}\\left(x\\right)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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[latex]x={3}^{3}=27[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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30. [latex]{\\text{log}}_{2}\\left(x\\right)=6[\/latex]<\/p>\n<\/div>\n<\/div>\n

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31. [latex]{\\text{log}}_{9}\\left(x\\right)=\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n

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[latex]x={9}^{\\frac{1}{2}}=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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32. [latex]{\\text{log}}_{18}\\left(x\\right)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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[latex]x={6}^{-3}=\\frac{1}{216}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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34. [latex]\\text{log}\\left(x\\right)=3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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

For the following exercises, use the definition of common and natural logarithms to simplify.<\/p>\n

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36. [latex]\\text{log}\\left({100}^{8}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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37. [latex]{10}^{\\text{log}\\left(32\\right)}[\/latex]<\/p>\n<\/div>\n

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[latex]32[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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38. [latex]2\\text{log}\\left(.0001\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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39. [latex]{e}^{\\mathrm{ln}\\left(1.06\\right)}[\/latex]<\/p>\n<\/div>\n

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[latex]1.06[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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40. [latex]\\mathrm{ln}\\left({e}^{-5.03}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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41. [latex]{e}^{\\mathrm{ln}\\left(10.125\\right)}+4[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]14.125[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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 <\/p>\n

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

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

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[latex]{\\mathrm{log}}_{b}\\left(2\\right)+{\\mathrm{log}}_{b}\\left(7\\right)+{\\mathrm{log}}_{b}\\left(x\\right)+{\\mathrm{log}}_{b}\\left(y\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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

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

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

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[latex]-k\\mathrm{ln}\\left(4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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

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[latex]\\mathrm{ln}\\left(7xy\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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49. [latex]{\\mathrm{log}}_{3}\\left(2\\right)+{\\mathrm{log}}_{3}\\left(a\\right)+{\\mathrm{log}}_{3}\\left(11\\right)+{\\mathrm{log}}_{3}\\left(b\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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[latex]{\\mathrm{log}}_{b}\\left(4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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

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[latex]{\\text{log}}_{b}\\left(7\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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Numeric<\/h4>\n

For the following exercises, evaluate the base[latex]\\text{ }b\\text{ }[\/latex]logarithmic expression without using a calculator.<\/p>\n

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54. [latex]{\\text{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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55. [latex]{\\text{log}}_{6}\\left(\\sqrt{6}\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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56. [latex]{\\text{log}}_{2}\\left(\\frac{1}{8}\\right)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n

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57. [latex]6{\\text{log}}_{8}\\left(4\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, evaluate the common logarithmic expression without using a calculator.<\/p>\n

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58. [latex]\\text{log}\\left(10,000\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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59. [latex]\\text{log}\\left(0.001\\right)[\/latex]<\/p>\n<\/div>\n

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60. [latex]-\\text{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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61. [latex]\\text{log}\\left(1\\right)+7[\/latex]<\/p>\n<\/div>\n<\/div>\n

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62. [latex]2\\text{log}\\left({100}^{-3}\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]-12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, evaluate the natural logarithmic expression without using a calculator.<\/p>\n

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63. [latex]\\text{ln}\\left({e}^{\\frac{1}{3}}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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64. [latex]\\text{ln}\\left(1\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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65. [latex]\\text{ln}\\left({e}^{-0.225}\\right)-3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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66. [latex]25\\text{ln}\\left({e}^{\\frac{2}{5}}\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Technology<\/h4>\n

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.<\/p>\n

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67. [latex]\\text{log}\\left(0.04\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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68. [latex]\\text{ln}\\left(15\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{2}.\\text{7}0\\text{8}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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69. [latex]\\text{ln}\\left(\\frac{4}{5}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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70. [latex]\\text{log}\\left(\\sqrt{2}\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]0.151[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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71. [latex]\\text{ln}\\left(\\sqrt{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Extensions<\/h4>\n
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72. Is[latex]\\text{ }x=0\\text{ }[\/latex]in the domain of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(x\\right)?\\text{ }[\/latex]If so, what is the value of the function when[latex]\\text{ }x=0?\\text{ }[\/latex]Verify the result.<\/p>\n<\/div>\n

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No, the function has no defined value for[latex]\\text{ }x=0.\\text{ }[\/latex]To verify, suppose[latex]\\text{ }x=0\\text{ }[\/latex]is in the domain of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(x\\right).\\text{ }[\/latex]Then there is some number[latex]\\text{ }n\\text{ }[\/latex]such that[latex]\\text{ }n=\\mathrm{log}\\left(0\\right).\\text{ }[\/latex]Rewriting as an exponential equation gives:[latex]\\text{ }{10}^{n}=0,[\/latex] which is impossible since no such real number[latex]\\text{ }n\\text{ }[\/latex]exists. Therefore,[latex]\\text{ }x=0\\text{ }[\/latex]is not<\/em> the domain of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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73. Is[latex]\\text{ }f\\left(x\\right)=0\\text{ }[\/latex]in the range of the function[latex]\\text{ }f\\left(x\\right)=\\mathrm{log}\\left(x\\right)?\\text{ }[\/latex]If so, for what value of[latex]\\text{ }x?\\text{ }[\/latex]Verify the result.<\/p>\n<\/div>\n<\/div>\n

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74. Is there a number[latex]\\text{ }x\\text{ }[\/latex]such that[latex]\\text{ }\\mathrm{ln}x=2?\\text{ }[\/latex]If so, what is that number? Verify the result.<\/p>\n<\/div>\n

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Yes. Suppose there exists a real number[latex]\\text{ }x\\text{ }[\/latex]such that[latex]\\text{ }\\mathrm{ln}x=2.\\text{ }[\/latex]Rewriting as an exponential equation gives[latex]\\text{ }x={e}^{2},[\/latex] which is a real number. To verify, let[latex]\\text{ }x={e}^{2}.\\text{ }[\/latex]Then, by definition,[latex]\\text{ }\\mathrm{ln}\\left(x\\right)=\\mathrm{ln}\\left({e}^{2}\\right)=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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75. Is the following true:[latex]\\text{ }\\frac{{\\mathrm{log}}_{3}\\left(27\\right)}{{\\mathrm{log}}_{4}\\left(\\frac{1}{64}\\right)}=-1?\\text{ }[\/latex]Verify the result.<\/p>\n<\/div>\n<\/div>\n

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76. Is the following true:[latex]\\text{ }\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}=1.725?\\text{ }[\/latex]Verify the result.<\/p>\n<\/div>\n

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No;[latex]\\text{ }\\mathrm{ln}\\left(1\\right)=0,[\/latex] so[latex]\\text{ }\\frac{\\mathrm{ln}\\left({e}^{1.725}\\right)}{\\mathrm{ln}\\left(1\\right)}\\text{ }[\/latex]is undefined.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Real-World Applications<\/h4>\n
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77. The exposure index[latex]\\text{ }EI\\text{ }[\/latex]for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation[latex]\\text{ }EI={\\mathrm{log}}_{2}\\left(\\frac{{f}^{2}}{t}\\right),[\/latex] where[latex]\\text{ }f\\text{ }[\/latex]is the \u201cf-stop\u201d setting on the camera, and [latex]t[\/latex] is the exposure time in seconds. Suppose the f-stop setting is[latex]\\text{ }8\\text{ }[\/latex]and the desired exposure time is[latex]\\text{ }2\\text{ }[\/latex]seconds. What will the resulting exposure index be?<\/p>\n<\/div>\n<\/div>\n

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78. Refer to the previous exercise. Suppose the light meter on a camera indicates an[latex]\\text{ }EI\\text{ }[\/latex]of[latex]\\text{ }-2,[\/latex] and the desired exposure time is 16 seconds. What should the f-stop setting be?<\/p>\n<\/div>\n

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

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79. The intensity levels I<\/em> of two earthquakes measured on a seismograph can be compared by the formula[latex]\\text{ }\\mathrm{log}\\frac{{I}_{1}}{{I}_{2}}={M}_{1}-{M}_{2}\\text{ }[\/latex]where[latex]\\text{ }M\\text{ }[\/latex]is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0[1]<\/sup><\/a>.\u00a0How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

  1. http:\/\/earthquake.usgs.gov\/earthquakes\/world\/historical.php. Accessed 3\/4\/2014. ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":158108,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2366","chapter","type-chapter","status-web-only","hentry"],"part":223,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2366","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/158108"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2366\/revisions"}],"predecessor-version":[{"id":2367,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2366\/revisions\/2367"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/223"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2366\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=2366"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2366"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=2366"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=2366"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}