Chapter 5

Exer. 1-4: Use the graph of \(y=e^{x}\) to help sketch the graph of \(f\). (a) \(f(x)=e^{-x}\) (b) \(f(x)=-e^{x}\)

Express in terms of logarithms of \(x, y, z\), or \(w\). (a) \(\log _{4}(x z)\) (b) \(\log _{4}(y / x)\) (c) \(\log _{4} \sqrt[3]{z}\)

Exer. 1-2: Change to logarithmic form. (a) \(4^{3}=64\) (b) \(4^{-3}=\frac{1}{64}\) (c) \(t^{r}=s\) (d) \(3^{x}=4-t\) (e) \(5^{7 t}=\frac{a+b}{a}\) (f) \((0.7)^{t}=5.3\)

Exer. 1-2: If possible, find (a) \(f^{-1}(5)\) and (b) \(g^{-1}(6)\) $$ \begin{aligned} &\begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & 2 & 4 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 3 & 5 & 9 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline x & 1 & 3 & 5 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & 6 & 2 & 6 \\ \hline \end{array} \end{aligned} $$

Solve the equation. $$7^{x+6}=7^{3 x-4}$$

Express in terms of logarithms of \(x, y, z\), or \(w\). (a) \(\log _{3}(x y z)\) (b) \(\log _{3}(x z / y)\) (c) \(\log _{3} \sqrt[5]{y}\)

Exer. 1-2: Change to logarithmic form. (a) \(3^{5}=243\) (b) \(3^{-4}=\frac{1}{81}\) (c) \(c^{p}=d\) (d) \(7^{x}=100 p\) (e) \(3^{-2 x}=\frac{P}{F}\) (f) \((0.9)^{t}=\frac{1}{2}\)

Exer. 1-2: If possible, find (a) \(f^{-1}(5)\) and (b) \(g^{-1}(6)\) $$ \begin{aligned} &\begin{array}{|l|l|l|l|} \hline t & 0 & 3 & 5 \\ \hline f(t) & 2 & 5 & 6 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|} \hline \boldsymbol{t} & 1 & 2 & 4 \\ \hline \boldsymbol{g}(t) & 3 & 6 & 6 \\ \hline \end{array} \end{aligned} $$

Solve the equation. $$6^{7-x}=6^{2 x+1}$$

Express in terms of logarithms of \(x, y, z\), or \(w\). $$ \log _{a} \frac{x^{3} w}{y^{2} z^{4}} $$

Use the graph of \(y=e^{x}\) to help sketch the graph of \(f\). (a) \(f(x)=e^{x+4}\) (b) \(f(x)=e^{x}+4\)

Exer. 3-4: Change to exponential form. (a) \(\log _{2} 32=5\) (b) \(\log _{3} \frac{1}{243}=-5\) (c) \(\log _{s} r=p\) (d) \(\log _{3}(x+2)=5\) (e) \(\log _{2} m=3 x+4\) (f) \(\log _{b} 512=\frac{3}{2}\)

Solve the equation. $$3^{2 x+3}=3^{\left(x^{2}\right)}$$

Express in terms of logarithms of \(x, y, z\), or \(w\). $$ \log _{6} \frac{y^{5} w^{2}}{x^{4} z^{3}} $$

Exer. 3-4: Change to exponential form. (a) \(\log _{3} 81=4\) (b) \(\log _{4} \frac{1}{256}=-4\) (c) \(\log _{v} w=q\) (d) \(\log _{6}(2 x-1)=3\) (e) \(\log _{4} p=5-x\) (f) \(\log _{a} 343=\frac{3}{4}\)

Solve the equation. $$9^{\left(x^{2}\right)}=3^{3 x+2}$$

Express in terms of logarithms of \(x, y, z\), or \(w\). $$ \log \frac{\sqrt{y}}{x^{4} \sqrt[3]{z}} $$

Exer. 5-8: Estimate using the change of base formula. $$ \log _{5} 6 $$

Exer. 5-10: Solve for \(t\) using logarithms with base \(a\). $$ 2 a^{t / 3}=5 $$

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=3 x-7 $$

Solve the equation. $$2^{-100 x}=(0.5)^{x-4}$$

Express in terms of logarithms of \(x, y, z\), or \(w\). $$ \log \frac{\sqrt{y}}{x^{4} \sqrt[3]{z}} $$

Exer. 5-8: Estimate using the change of base formula. $$ \log _{2} 20 $$

Exer. 5-10: Solve for \(t\) using logarithms with base \(a\). $$ 3 a^{4 t}=10 $$

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=\frac{1}{x-2} $$

Solve the equation. $$\left(\frac{1}{2}\right)^{6-x}=2$$

Express in terms of logarithms of \(x, y, z\), or \(w\). $$ \ln \sqrt[4]{\frac{x^{7}}{y^{5} z}} $$

How much money, invested at an interest rate of \(r \%\) per year compounded continuously, will amount to \(A\) dollars after \(t\) years? $$ A=100,000, \quad r=6.4, \quad t=18 $$

Exer. 5-8: Estimate using the change of base formula. $$ \log _{9} 0.2 $$

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=x^{2}-9 $$

Solve the equation. $$4^{x-3}=8^{4-x}$$

Express in terms of logarithms of \(x, y, z\), or \(w\). $$ \ln x \sqrt[3]{\frac{y^{4}}{z^{5}}} $$

How much money, invested at an interest rate of \(r \%\) per year compounded continuously, will amount to \(A\) dollars after \(t\) years? A=15,000, \quad r=5.5, \quad t=4

Exer. 5-8: Estimate using the change of base formula. $$ \log _{6} \frac{1}{2} $$

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=x^{2}+4 $$

Solve the equation. $$27^{x-1}=9^{2 x-3}$$

Write the expression as one logarithm. (a) \(\log _{3} x+\log _{3}(5 y)\) (b) \(\log _{3}(2 z)-\log _{3} x\) (c) \(5 \log _{3} y\)

Exer. 9-10: Evaluate using the change of base formula (without a calculator). \( \frac{\log _{5} 16}{\log _{5} 4}\)

Exer. 5-10: Solve for \(t\) using logarithms with base \(a\). $$ A=B a^{C t}+D $$

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=\sqrt{x} $$

Solve the equation. $$4^{x} \cdot\left(\frac{1}{2}\right)^{3-2 x}=8 \cdot\left(2^{x}\right)^{2}$$

Write the expression as one logarithm. (a) \(\log _{4}(3 z)+\log _{4} x\) (b) \(\log _{4} x-\log _{4}(7 y)\) (c) \(\frac{1}{3} \log _{4} w\)

Exer. 9-10: Evaluate using the change of base formula (without a calculator). \(\frac{\log _{7} 243}{\log _{7} 3}\)

Exer. 5-10: Solve for \(t\) using logarithms with base \(a\). $$ L=M a^{t / N}-P $$

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=\sqrt[3]{x} $$

Solve the equation. $$9^{2 x} \cdot\left(\frac{1}{3}\right)^{x+2}=27 \cdot\left(3^{x}\right)^{-2}$$

Write the expression as one logarithm. $$ 2 \log _{a} x+\frac{1}{3} \log _{a}(x-2)-5 \log _{a}(2 x+3) $$

$$ e^{\left(x^{2}\right)}=e^{7 x-12} $$

Exer. 11-24: Find the exact solution, using common logarithms, and a two- decimal-place approximation of each solution, when appropriate. $$ 3^{x+4}=2^{1-3 x} $$

Exer. 11-12: Change to logarithmic form. (a) \(10^{5}=100,000\) (b) \(10^{-3}=0.001\) (c) \(10^{x}=y+1\) (d) \(e^{7}=p\) (e) \(e^{2 t}=3-x\)