14.3: Exercises
- Page ID
- 49040
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Combine the terms and write your answer as one logarithm.
- \(3\ln(x)+\ln(y)\)
- \(\log(x)-\dfrac 2 3\log(y)\)
- \(\dfrac{1}{3}\log(x)-\log(y)+4\log(z)\)
- \(\log(xy^2z^3)-\log(x^4y^3z^2)\)
- \(\dfrac 1 4\ln(x)-\dfrac{1}{2}\ln(y)+\dfrac{2}{3}\ln(z)\)
- \(-\ln(x^2-1)+\ln(x-1)\)
- \(5\ln(x)+2\ln(x^4)-3\ln(x)\) & h) \(\log_5(a^2+10a+9)-\log_5(a+9)+2\)
- Answer
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- \(\ln \left(x^{3} \cdot y\right)\)
- \(\log \left(\dfrac{x}{y^{\frac{2}{3}}}\right)=\log \left(\dfrac{x}{\sqrt[3]{y^{2}}}\right)\)
- \(\log \left(\dfrac{\sqrt[3]{x} z^{4}}{y}\right)\)
- \(\log \left(\dfrac{z}{x^{3} y}\right)\)
- \(\ln \left(\dfrac{\sqrt[4]{x} \sqrt[3]{z^{2}}}{\sqrt{y}}\right)\)
- \(\ln \left(\dfrac{1}{x+1}\right)\)
- \(\ln \left(x^{10}\right)\)
- \(\log _{5}(25 \cdot(a+1))\)
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 \(x,y,z>0\).
- \(\log(x^3\cdot y)\)
- \(\log(\sqrt[3]{x^2}\cdot \sqrt[4]{y^7})\)
- \(\log\left(\sqrt{x\cdot \sqrt[3]{y}}\right)\)
- \(\ln\left(\dfrac{x^3} {y^4}\right)\)
- \(\ln\left(\dfrac{x^2} {\sqrt{y}\cdot z^2}\right)\)
- \(\log_3\left(\sqrt{\dfrac{x\cdot y^3} {\sqrt{z}}}\,\right)\)
- \(\log_2\left(\dfrac{\sqrt[4]{x^3\cdot z}} {y^3}\right)\)
- \(\log\left(\dfrac{100 \sqrt[5]{z}}{y^2}\right)\)
- \(\ln \left(\sqrt[3]{\dfrac{\sqrt{y}\cdot z^4}{e^2}}\right)\)
- Answer
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- \(3 u+v\)
- \(\dfrac{2}{3} u+\dfrac{7}{4} v\)
- \(\dfrac{1}{2} u+\dfrac{1}{6} v\)
- \(3 u-4 v\)
- \(2 u-\dfrac{1}{2} v-2 w\)
- \(\dfrac{1}{2} u+\dfrac{3}{2} v-\dfrac{1}{4} w\)
- \(\dfrac{3}{4} u-3 v+\dfrac{1}{4} w\)
- \(2-2 v+\dfrac{1}{5} w\)
- \(\dfrac{1}{6} v+\dfrac{4}{3} w-\dfrac{2}{3}\)
Solve for \(x\) without using a calculator.
- \(6^{x-2}=36\)
- \(2^{3x-8}=16\)
- \(10^{5-x}=0.0001\) & d)
- \(5^{5x+7}=\dfrac{1}{125}\)
- \(2^x=64^{x+1}\)
- \(4^{x+3}=32^{x}\)
- \(13^{4+2x}=1\)
- \(3^{x+2}=27^{x-3}\)
- \(25^{7x-4}=5^{2-3x}\)
- \(9^{5+3x}=27^{8-2x}\)
- Answer
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- \(x = 4\)
- \(x = 4\)
- \(x = 9\)
- \(x = −2\)
- \(x = −\dfrac 6 5\)
- \(x = 2\)
- \(x = −2\)
- \(x = \dfrac {11}{2}\)
- \(x =\dfrac {10}{17}\)
- \(x = \dfrac 7 6\)
Solve for \(x\) without using a calculator.
- \(\ln(2x+4)=\ln(5x-5)\) & b)
- \(\ln(x+6)=\ln(x-2)+\ln(3)\)
- \(\log_2(x+5)=\log_2(x)+5\)
- \(\log(x)+1=\log(5x+380)\)
- \(\log(x+5)+\log(x)=\log(6)\)
- \(\log_2(x)+\log_2(x-6)=4\)
- \(\log_6(x)+\log_6(x-16)=2\)
- \(\log_5(x-24)+\log_5(x)=2\)
- \(\log_4(x)+\log_4(x+6)=2\)
- \(\log_2(x+3)+\log_2(x+5)=3\)
- Answer
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- \(x = 3\)
- \(x = 6\)
- \(x = \dfrac{5}{31}\)
- \(x = 76\)
- \(x = 1\)
- \(x = 8\)
- \(x = 18\)
- \(x = 25\)
- \(x = 2\)
- \(x = −1\)
Solve for \(x\). First find the exact answer as an expression involving logarithms. Then approximate the answer to the nearest hundredth using the calculator.
- \(4^{x}=57\)
- \(9^{x-2}=7\)
- \(2^{x+1}=31\)
- \(3.8^{2x+7}=63\)
- \(5^{x+5}=8^x\)
- \(3^{x+2}=0.4^x\)
- \(1.02^{2x-9}=4.35^{x}\)
- \(4^{x+1}=5^{x+2}\)
- \(9^{3-x}=4^{x-6}\)
- \(2.4^{7-2x}=3.8^{3x+4}\)
- \(4^{9x-2}=9^{2x-4}\)
- \(1.95^{-3x-4}=1.2^{4-7x}\)
- Answer
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- \(x=\dfrac{\log 57}{\log 4} \approx 2.92\)
- \(x=\dfrac{\log 7}{\log 9}+2 \approx 2.89\)
- \(x=\dfrac{\log 31}{\log 2}-1 \approx 3.94\)
- \(x=\dfrac{\log (63)-7 \log (3.8)}{2 \log (3.8)} \approx-1.95\)
- \(x=\dfrac{5 \cdot \log (5)}{\log (8)-\log (5)} \approx 17.12\)
- \(x=\dfrac{2 \cdot \log (3)}{\log (0.4)-\log (3)} \approx-1.09\)
- \(x=\dfrac{9 \log (1.02)}{2 \log (1.02)-\log (4.35)} \approx-0.12\)
- \(x=\dfrac{\log (4)-2 \log (5)}{\log (5)-\log (4)} \approx-8.21\)
- \(x=\dfrac{3 \log (9)+6 \log (4)}{\log (9)+\log (4)} \approx 4.16\)
- \(x=\dfrac{7 \log (2.4)-4 \log (3.8)}{2 \log (2.4)+3 \log (3.8)} \approx 0.14\)
- \(x=\dfrac{4 \log (9)-2 \log (4)}{2 \log (9)-9 \log (4)} \approx-0.74\)
- \(\dfrac{4 \log (1.2)+4 \log (1.95)}{7 \log (1.2)-3 \log (1.95)} \approx-4.68\)