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10.5E: 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.

      1. \(\log _{4} 1\)
      2. \(\log _{8} 8\)
      1. \(\log _{12} 1\)
      2. \(\ln e\)
      1. \(3^{\log _{3} 6}\)
      2. \(\log _{2} 2^{7}\)
      1. \(5^{\log _{5} 10}\)
      2. \(\log _{4} 4^{10}\)
      1. \(8^{\log _{8} 7}\)
      2. \(\log _{6} 6^{-2}\)
      1. \(6^{\log _{6} 15}\)
      2. \(\log _{8} 8^{-4}\)
      1. \(10^{\log \sqrt{5}}\)
      2. \(\log 10^{-2}\)
      1. \(10^{\log \sqrt{3}}\)
      2. \(\log 10^{-1}\)
      1. \(e^{\ln 4}\)
      2. \(\ln e^{2}\)
      1. \(e^{\ln 3}\)
      2. \(\ln e^{7}\)
    Answer

    2.

    1. \(0\)
    2. \(1\)

    4.

    1. \(10\)
    2. \(10\)

    6.

    1. \(15\)
    2. \(-4\)

    8.

    1. \(\sqrt{3}\)
    2. \(-1\)

    10.

    1. \(3\)
    2. \(7\)
    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.

    1. \(\log _{4} 6 x\)
    2. \(\log _{5} 8 y\)
    3. \(\log _{2} 32 x y\)
    4. \(\log _{3} 81 x y\)
    5. \(\log 100 x\)
    6. \(\log 1000 y\)
    Answer

    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.

    1. \(\log _{3} \frac{3}{8}\)
    2. \(\log _{6} \frac{5}{6}\)
    3. \(\log _{4} \frac{16}{y}\)
    4. \(\log _{5} \frac{125}{x}\)
    5. \(\log \frac{x}{10}\)
    6. \(\log \frac{10,000}{y}\)
    7. \(\ln \frac{e^{3}}{3}\)
    8. \(\ln \frac{e^{4}}{16}\)
    Answer

    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.

    1. \(\log _{3} x^{2}\)
    2. \(\log _{2} x^{5}\)
    3. \(\log x^{-2}\)
    4. \(\log x^{-3}\)
    5. \(\log _{4} \sqrt{x}\)
    6. \(\log _{5} \sqrt[3]{x}\)
    7. \(\ln x^{\sqrt{3}}\)
    8. \(\ln x^{\sqrt[3]{4}}\)
    Answer

    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.

    1. \(\log _{5}\left(4 x^{6} y^{4}\right)\)
    2. \(\log _{2}\left(3 x^{5} y^{3}\right)\)
    3. \(\log _{3}\left(\sqrt{2} x^{2}\right)\)
    4. \(\log _{5}\left(\sqrt[4]{21} y^{3}\right)\)
    5. \(\log _{3} \frac{x y^{2}}{z^{2}}\)
    6. \(\log _{5} \frac{4 a b^{3} c^{4}}{d^{2}}\)
    7. \(\log _{4} \frac{\sqrt{x}}{16 y^{4}}\)
    8. \(\log _{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}\)
    9. \(\log _{2} \frac{\sqrt{2 x+y^{2}}}{z^{2}}\)
    10. \(\log _{3} \frac{\sqrt{3 x+2 y^{2}}}{5 z^{2}}\)
    11. \(\log _{2} \sqrt[4]{\frac{5 x^{3}}{2 y^{2} z^{4}}}\)
    12. \(\log _{5} \sqrt[3]{\frac{3 x^{2}}{4 y^{3} z}}\)
    Answer

    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.

    1. \(\log _{6} 4+\log _{6} 9\)
    2. \(\log 4+\log 25\)
    3. \(\log _{2} 80-\log _{2} 5\)
    4. \(\log _{3} 36-\log _{3} 4\)
    5. \(\log _{3} 4+\log _{3}(x+1)\)
    6. \(\log _{2} 5-\log _{2}(x-1)\)
    7. \(\log _{7} 3+\log _{7} x-\log _{7} y\)
    8. \(\log _{5} 2-\log _{5} x-\log _{5} y\)
    9. \(4 \log _{2} x+6 \log _{2} y\)
    10. \(6 \log _{3} x+9 \log _{3} y\)
    11. \(\log _{3}\left(x^{2}-1\right)-2 \log _{3}(x-1)\)
    12. \(\log \left(x^{2}+2 x+1\right)-2 \log (x+1)\)
    13. \(4 \log x-2 \log y-3 \log z\)
    14. \(3 \ln x+4 \ln y-2 \ln z\)
    15. \(\frac{1}{3} \log x-3 \log (x+1)\)
    16. \(2 \log (2 x+3)+\frac{1}{2} \log (x+1)\)
    Answer

    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.

    1. \(\log _{3} 42\)
    2. \(\log _{5} 46\)
    3. \(\log _{12} 87\)
    4. \(\log _{15} 93\)
    5. \(\log _{\sqrt{2}} 17\)
    6. \(\log _{\sqrt{3}} 21\)
    Answer

    2. \(2.379\)

    4. \(1.674\)

    6. \(5.542\)

    Exercise \(\PageIndex{28}\) Writing Exercises
    1. 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?
    2. 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?
    3. Use an example to show that \(\log (a+b) \neq \log a+\log b ?\)
    4. Explain how to find the value of \(\log _{7} 15\) using your calculator.
    Answer

    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.

    This table has three rows and four columns. The first row, which serves as a header, reads I can…, Confidently, With some help, and No—I don’t get it. The first column below the header row reads use the properties of logarithms and use the change of base formula. The rest of the cells are blank.
    Figure 10.4.5

    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?

    This page titled 10.5E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.