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14.3: Exercises

Last updated

May 2, 2022
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- 14.2: Solving Exponential and Logarithmic Equations
- 15: Applications Exponentials and Logarithms

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Exercise $\PageIndex{1}$

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

$\ln \left(x^{3} \cdot y\right)$
$\log \left(\dfrac{x}{y^{\frac{2}{3}}}\right)=\log \left(\dfrac{x}{\sqrt[3]{y^{2}}}\right)$
$\log \left(\dfrac{\sqrt[3]{x} z^{4}}{y}\right)$
$\log \left(\dfrac{z}{x^{3} y}\right)$
$\ln \left(\dfrac{\sqrt[4]{x} \sqrt[3]{z^{2}}}{\sqrt{y}}\right)$
$\ln \left(\dfrac{1}{x+1}\right)$
$\ln \left(x^{10}\right)$
$\log _{5}(25 \cdot(a+1))$

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 $x,y,z>0$ .

$\log(x^3\cdot y)$
$\log(\sqrt[3]{x^2}\cdot \sqrt[4]{y^7})$
$\log\left(\sqrt{x\cdot \sqrt[3]{y}}\right)$
$\ln\left(\dfrac{x^3} {y^4}\right)$
$\ln\left(\dfrac{x^2} {\sqrt{y}\cdot z^2}\right)$
$\log_3\left(\sqrt{\dfrac{x\cdot y^3} {\sqrt{z}}}\,\right)$
$\log_2\left(\dfrac{\sqrt[4]{x^3\cdot z}} {y^3}\right)$
$\log\left(\dfrac{100 \sqrt[5]{z}}{y^2}\right)$
$\ln \left(\sqrt[3]{\dfrac{\sqrt{y}\cdot z^4}{e^2}}\right)$

Answer

$3 u+v$
$\dfrac{2}{3} u+\dfrac{7}{4} v$
$\dfrac{1}{2} u+\dfrac{1}{6} v$
$3 u-4 v$
$2 u-\dfrac{1}{2} v-2 w$
$\dfrac{1}{2} u+\dfrac{3}{2} v-\dfrac{1}{4} w$
$\dfrac{3}{4} u-3 v+\dfrac{1}{4} w$
$2-2 v+\dfrac{1}{5} w$
$\dfrac{1}{6} v+\dfrac{4}{3} w-\dfrac{2}{3}$

Exercise $\PageIndex{3}$

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

$x = 4$
$x = 4$
$x = 9$
$x = −2$
$x = −\dfrac 6 5$
$x = 2$
$x = −2$
$x = \dfrac {11}{2}$
$x =\dfrac {10}{17}$
$x = \dfrac 7 6$

Exercise $\PageIndex{4}$

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

$x = 3$
$x = 6$
$x = \dfrac{5}{31}$
$x = 76$
$x = 1$
$x = 8$
$x = 18$
$x = 25$
$x = 2$
$x = −1$

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.

$4^{x}=57$
$9^{x-2}=7$
$2^{x+1}=31$
$3.8^{2x+7}=63$
$5^{x+5}=8^x$
$3^{x+2}=0.4^x$
$1.02^{2x-9}=4.35^{x}$
$4^{x+1}=5^{x+2}$
$9^{3-x}=4^{x-6}$
$2.4^{7-2x}=3.8^{3x+4}$
$4^{9x-2}=9^{2x-4}$
$1.95^{-3x-4}=1.2^{4-7x}$

Answer

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

Exercise 14.3.1\PageIndex{1}

Exercise 14.3.2\PageIndex{2}

Exercise 14.3.3\PageIndex{3}

Exercise 14.3.4\PageIndex{4}

Exercise 14.3.5\PageIndex{5}

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Exercise $\PageIndex{1}$

Exercise $\PageIndex{2}$

Exercise $\PageIndex{3}$

Exercise $\PageIndex{4}$

Exercise $\PageIndex{5}$