12.5E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{21}\) Use the Properties of Logarithms
In the following exercises, use the properties of logarithms to evaluate.
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- \(\log _{4} 1\)
- \(\log _{8} 8\)
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- \(\log _{12} 1\)
- \(\ln e\)
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- \(3^{\log _{3} 6}\)
- \(\log _{2} 2^{7}\)
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- \(5^{\log _{5} 10}\)
- \(\log _{4} 4^{10}\)
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- \(8^{\log _{8} 7}\)
- \(\log _{6} 6^{-2}\)
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- \(6^{\log _{6} 15}\)
- \(\log _{8} 8^{-4}\)
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- \(10^{\log \sqrt{5}}\)
- \(\log 10^{-2}\)
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- \(10^{\log \sqrt{3}}\)
- \(\log 10^{-1}\)
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- \(e^{\ln 4}\)
- \(\ln e^{2}\)
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- \(e^{\ln 3}\)
- \(\ln e^{7}\)
- Answer
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2.
- \(0\)
- \(1\)
4.
- \(10\)
- \(10\)
6.
- \(15\)
- \(-4\)
8.
- \(\sqrt{3}\)
- \(-1\)
10.
- \(3\)
- \(7\)
Exercise \(\PageIndex{22}\) Use the Properties of Logarithms
In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
- \(\log _{4} 6 x\)
- \(\log _{5} 8 y\)
- \(\log _{2} 32 x y\)
- \(\log _{3} 81 x y\)
- \(\log 100 x\)
- \(\log 1000 y\)
- Answer
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2. \(\log _{5} 8+\log _{5} y\)
4. \(4+\log _{3} x+\log _{3} y\)
6. \(3+\log y\)
Exercise \(\PageIndex{23}\) Use the Properties of Logarithms
In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
- \(\log _{3} \frac{3}{8}\)
- \(\log _{6} \frac{5}{6}\)
- \(\log _{4} \frac{16}{y}\)
- \(\log _{5} \frac{125}{x}\)
- \(\log \frac{x}{10}\)
- \(\log \frac{10,000}{y}\)
- \(\ln \frac{e^{3}}{3}\)
- \(\ln \frac{e^{4}}{16}\)
- Answer
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2. \(\log _{6} 5-1\)
4. \(3-\log _{5} x\)
6. \(4-\log y\)
8. \(4-\ln 16\)
Exercise \(\PageIndex{24}\) Use the Properties of Logarithms
In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.
- \(\log _{3} x^{2}\)
- \(\log _{2} x^{5}\)
- \(\log x^{-2}\)
- \(\log x^{-3}\)
- \(\log _{4} \sqrt{x}\)
- \(\log _{5} \sqrt[3]{x}\)
- \(\ln x^{\sqrt{3}}\)
- \(\ln x^{\sqrt[3]{4}}\)
- Answer
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2. \(5\log _{2} x\)
4. \(-3 \log x\)
6. \(\frac{1}{3} \log _{5} x\)
8. \(\sqrt[3]{4} \ln x\)
Exercise \(\PageIndex{25}\) Use the Properties of Logarithms
In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.
- \(\log _{5}\left(4 x^{6} y^{4}\right)\)
- \(\log _{2}\left(3 x^{5} y^{3}\right)\)
- \(\log _{3}\left(\sqrt{2} x^{2}\right)\)
- \(\log _{5}\left(\sqrt[4]{21} y^{3}\right)\)
- \(\log _{3} \frac{x y^{2}}{z^{2}}\)
- \(\log _{5} \frac{4 a b^{3} c^{4}}{d^{2}}\)
- \(\log _{4} \frac{\sqrt{x}}{16 y^{4}}\)
- \(\log _{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}\)
- \(\log _{2} \frac{\sqrt{2 x+y^{2}}}{z^{2}}\)
- \(\log _{3} \frac{\sqrt{3 x+2 y^{2}}}{5 z^{2}}\)
- \(\log _{2} \sqrt[4]{\frac{5 x^{3}}{2 y^{2} z^{4}}}\)
- \(\log _{5} \sqrt[3]{\frac{3 x^{2}}{4 y^{3} z}}\)
- Answer
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2. \(\log _{2} 3+5 \log _{2} x+3 \log _{2} y\)
4. \(\frac{1}{4} \log _{5} 21+3 \log _{5} y\)
6. \(\begin{array}{l}{\log _{5} 4+\log _{5} a+3 \log _{5} b} {+4 \log _{5} c-2 \log _{5} d}\end{array}\)
8. \(\frac{2}{3} \log _{3} x-3-4 \log _{3} y\)
10. \(\frac{1}{2} \log _{3}\left(3 x+2 y^{2}\right)-\log _{3} 5-2 \log _{3} z\)
12. \(\begin{array}{l}{\frac{1}{3}\left(\log _{5} 3+2 \log _{5} x-\log _{5} 4\right.} {-3 \log _{5} y-\log _{5} z )}\end{array}\)
Exercise \(\PageIndex{26}\) Use the Properties of Logarithms
In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
- \(\log _{6} 4+\log _{6} 9\)
- \(\log 4+\log 25\)
- \(\log _{2} 80-\log _{2} 5\)
- \(\log _{3} 36-\log _{3} 4\)
- \(\log _{3} 4+\log _{3}(x+1)\)
- \(\log _{2} 5-\log _{2}(x-1)\)
- \(\log _{7} 3+\log _{7} x-\log _{7} y\)
- \(\log _{5} 2-\log _{5} x-\log _{5} y\)
- \(4 \log _{2} x+6 \log _{2} y\)
- \(6 \log _{3} x+9 \log _{3} y\)
- \(\log _{3}\left(x^{2}-1\right)-2 \log _{3}(x-1)\)
- \(\log \left(x^{2}+2 x+1\right)-2 \log (x+1)\)
- \(4 \log x-2 \log y-3 \log z\)
- \(3 \ln x+4 \ln y-2 \ln z\)
- \(\frac{1}{3} \log x-3 \log (x+1)\)
- \(2 \log (2 x+3)+\frac{1}{2} \log (x+1)\)
- Answer
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2. \(2\)
4. \(2\)
6. \(\log _{2} \frac{5}{x-1}\)
8. \(\log _{5} \frac{2}{x y}\)
10. \(\log _{3} x^{6} y^{9}\)
12. \(0\)
14. \(\ln \frac{x^{3} y^{4}}{z^{2}}\)
16. \(\log (2 x+3)^{2} \cdot \sqrt{x+1}\)
Exercise \(\PageIndex{27}\) Use the Change-of-Base Formula
In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.
- \(\log _{3} 42\)
- \(\log _{5} 46\)
- \(\log _{12} 87\)
- \(\log _{15} 93\)
- \(\log _{\sqrt{2}} 17\)
- \(\log _{\sqrt{3}} 21\)
- Answer
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2. \(2.379\)
4. \(1.674\)
6. \(5.542\)
Exercise \(\PageIndex{28}\) Writing Exercises
- Write the Product Property in your own words. Does it apply to each of the following? \(\log _{a} 5 x, \log _{a}(5+x)\). Why or why not?
- Write the Power Property in your own words. Does it apply to each of the following? \(\log _{a} x^{p},\left(\log _{a} x\right)^{r}\). Why or why not?
- Use an example to show that \(\log (a+b) \neq \log a+\log b ?\)
- Explain how to find the value of \(\log _{7} 15\) using your calculator.
- Answer
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2. Answers may vary
4. Answers may vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. On a scale of \(1−10\), how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?