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14.3: Exercises

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    Exercise \(\PageIndex{1}\)

    Combine the terms and write your answer as one logarithm.

    1. \(3\ln(x)+\ln(y)\)
    2. \(\log(x)-\dfrac 2 3\log(y)\)
    3. \(\dfrac{1}{3}\log(x)-\log(y)+4\log(z)\)
    4. \(\log(xy^2z^3)-\log(x^4y^3z^2)\)
    5. \(\dfrac 1 4\ln(x)-\dfrac{1}{2}\ln(y)+\dfrac{2}{3}\ln(z)\)
    6. \(-\ln(x^2-1)+\ln(x-1)\)
    7. \(5\ln(x)+2\ln(x^4)-3\ln(x)\) & h) \(\log_5(a^2+10a+9)-\log_5(a+9)+2\)
    Answer
    1. \(\ln \left(x^{3} \cdot y\right)\)
    2. \(\log \left(\dfrac{x}{y^{\frac{2}{3}}}\right)=\log \left(\dfrac{x}{\sqrt[3]{y^{2}}}\right)\)
    3. \(\log \left(\dfrac{\sqrt[3]{x} z^{4}}{y}\right)\)
    4. \(\log \left(\dfrac{z}{x^{3} y}\right)\)
    5. \(\ln \left(\dfrac{\sqrt[4]{x} \sqrt[3]{z^{2}}}{\sqrt{y}}\right)\)
    6. \(\ln \left(\dfrac{1}{x+1}\right)\)
    7. \(\ln \left(x^{10}\right)\)
    8. \(\log _{5}(25 \cdot(a+1))\)

    Exercise \(\PageIndex{2}\)

    Write the expressions in terms of elementary logarithms \(u=\log_b(x)\), \(v=\log_b(y)\), and \(w=\log_b(z)\) (whichever are applicable). Assume that \(x,y,z>0\).

    1. \(\log(x^3\cdot y)\)
    2. \(\log(\sqrt[3]{x^2}\cdot \sqrt[4]{y^7})\)
    3. \(\log\left(\sqrt{x\cdot \sqrt[3]{y}}\right)\)
    4. \(\ln\left(\dfrac{x^3} {y^4}\right)\)
    5. \(\ln\left(\dfrac{x^2} {\sqrt{y}\cdot z^2}\right)\)
    6. \(\log_3\left(\sqrt{\dfrac{x\cdot y^3} {\sqrt{z}}}\,\right)\)
    7. \(\log_2\left(\dfrac{\sqrt[4]{x^3\cdot z}} {y^3}\right)\)
    8. \(\log\left(\dfrac{100 \sqrt[5]{z}}{y^2}\right)\)
    9. \(\ln \left(\sqrt[3]{\dfrac{\sqrt{y}\cdot z^4}{e^2}}\right)\)
    Answer
    1. \(3 u+v\)
    2. \(\dfrac{2}{3} u+\dfrac{7}{4} v\)
    3. \(\dfrac{1}{2} u+\dfrac{1}{6} v\)
    4. \(3 u-4 v\)
    5. \(2 u-\dfrac{1}{2} v-2 w\)
    6. \(\dfrac{1}{2} u+\dfrac{3}{2} v-\dfrac{1}{4} w\)
    7. \(\dfrac{3}{4} u-3 v+\dfrac{1}{4} w\)
    8. \(2-2 v+\dfrac{1}{5} w\)
    9. \(\dfrac{1}{6} v+\dfrac{4}{3} w-\dfrac{2}{3}\)

    Exercise \(\PageIndex{3}\)

    Solve for \(x\) without using a calculator.

    1. \(6^{x-2}=36\)
    2. \(2^{3x-8}=16\)
    3. \(10^{5-x}=0.0001\) & d)
    4. \(5^{5x+7}=\dfrac{1}{125}\)
    5. \(2^x=64^{x+1}\)
    6. \(4^{x+3}=32^{x}\)
    7. \(13^{4+2x}=1\)
    8. \(3^{x+2}=27^{x-3}\)
    9. \(25^{7x-4}=5^{2-3x}\)
    10. \(9^{5+3x}=27^{8-2x}\)
    Answer
    1. \(x = 4\)
    2. \(x = 4\)
    3. \(x = 9\)
    4. \(x = −2\)
    5. \(x = −\dfrac 6 5\)
    6. \(x = 2\)
    7. \(x = −2\)
    8. \(x = \dfrac {11}{2}\)
    9. \(x =\dfrac {10}{17}\)
    10. \(x = \dfrac 7 6\)

    Exercise \(\PageIndex{4}\)

    Solve for \(x\) without using a calculator.

    1. \(\ln(2x+4)=\ln(5x-5)\) & b)
    2. \(\ln(x+6)=\ln(x-2)+\ln(3)\)
    3. \(\log_2(x+5)=\log_2(x)+5\)
    4. \(\log(x)+1=\log(5x+380)\)
    5. \(\log(x+5)+\log(x)=\log(6)\)
    6. \(\log_2(x)+\log_2(x-6)=4\)
    7. \(\log_6(x)+\log_6(x-16)=2\)
    8. \(\log_5(x-24)+\log_5(x)=2\)
    9. \(\log_4(x)+\log_4(x+6)=2\)
    10. \(\log_2(x+3)+\log_2(x+5)=3\)
    Answer
    1. \(x = 3\)
    2. \(x = 6\)
    3. \(x = \dfrac{5}{31}\)
    4. \(x = 76\)
    5. \(x = 1\)
    6. \(x = 8\)
    7. \(x = 18\)
    8. \(x = 25\)
    9. \(x = 2\)
    10. \(x = −1\)

    Exercise \(\PageIndex{5}\)

    Solve for \(x\). First find the exact answer as an expression involving logarithms. Then approximate the answer to the nearest hundredth using the calculator.

    1. \(4^{x}=57\)
    2. \(9^{x-2}=7\)
    3. \(2^{x+1}=31\)
    4. \(3.8^{2x+7}=63\)
    5. \(5^{x+5}=8^x\)
    6. \(3^{x+2}=0.4^x\)
    7. \(1.02^{2x-9}=4.35^{x}\)
    8. \(4^{x+1}=5^{x+2}\)
    9. \(9^{3-x}=4^{x-6}\)
    10. \(2.4^{7-2x}=3.8^{3x+4}\)
    11. \(4^{9x-2}=9^{2x-4}\)
    12. \(1.95^{-3x-4}=1.2^{4-7x}\)
    Answer
    1. \(x=\dfrac{\log 57}{\log 4} \approx 2.92\)
    2. \(x=\dfrac{\log 7}{\log 9}+2 \approx 2.89\)
    3. \(x=\dfrac{\log 31}{\log 2}-1 \approx 3.94\)
    4. \(x=\dfrac{\log (63)-7 \log (3.8)}{2 \log (3.8)} \approx-1.95\)
    5. \(x=\dfrac{5 \cdot \log (5)}{\log (8)-\log (5)} \approx 17.12\)
    6. \(x=\dfrac{2 \cdot \log (3)}{\log (0.4)-\log (3)} \approx-1.09\)
    7. \(x=\dfrac{9 \log (1.02)}{2 \log (1.02)-\log (4.35)} \approx-0.12\)
    8. \(x=\dfrac{\log (4)-2 \log (5)}{\log (5)-\log (4)} \approx-8.21\)
    9. \(x=\dfrac{3 \log (9)+6 \log (4)}{\log (9)+\log (4)} \approx 4.16\)
    10. \(x=\dfrac{7 \log (2.4)-4 \log (3.8)}{2 \log (2.4)+3 \log (3.8)} \approx 0.14\)
    11. \(x=\dfrac{4 \log (9)-2 \log (4)}{2 \log (9)-9 \log (4)} \approx-0.74\)
    12. \(\dfrac{4 \log (1.2)+4 \log (1.95)}{7 \log (1.2)-3 \log (1.95)} \approx-4.68\)

    This page titled 14.3: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.