Chapter 4

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

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\text { (a) } \log _{4}(x z)$$ (b) \(\log _{4}(y / x)\) (c) \(\log _{4} \sqrt[3]{z}\)

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

Find \(\begin{array}{ll}\text { (a) } f^{-1}(5) & \text { (b) } g^{-1}(6)\end{array}\) $$\begin{array}{|c|c|c|c|} \hline x & 2 & 4 & 6 \\ \hline f(x) & 3 & 5 & 9 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|} \hline x & 1 & 3 & 5 \\ \hline g(x) & 6 & 2 & 6 \\ \hline \end{array}$$

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

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) (a) \(\log _{3}(x y z)\) (b) \(\log _{3}(x z / y)\) (c) \(\log _{3} \sqrt[5]{y}\)

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

Find \(\begin{array}{ll}\text { (a) } f^{-1}(5) & \text { (b) } g^{-1}(6)\end{array}\) $$\begin{array}{|l|l|ll|} \hline t & 0 & 3 & 5 \\ \hline f(t) & 2 & 5 & 6 \\ \hline \end{array}$$ $$\begin{array}{|c|c|c|c|} \hline t & 1 & 2 & 4 \\ \hline g(t) & 3 & 6 & 6 \\ \hline \end{array}$$

Use the graph of \(y=e^{x}\) to help sketch the graph of \(f\). (a) \(f(x)=e^{2 x}\) (b) \(f(x)=2 e^{x}\)

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

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\log _{x} \frac{x^{3} w}{y^{2} z^{4}}$$

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

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

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

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\log _{a} \frac{y^{5} w^{2}}{x^{4} z^{3}}$$

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

Use the graph of \(y=e^{x}\) to help sketch the graph of \(f\). (a) \(f(x)=e^{-2 x}\) (b) \(f(x)=-2 e^{x}\)

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

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\log \frac{\sqrt[3]{z}}{x \sqrt{y}}$$

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

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

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

If \(P\) dollars is deposited in a savings account that pays interest at a rate of \(r \%\) per year compounded continuously, find the balance after \(t\) years. $$P=1000, \quad r=8 \frac{1}{4}, \quad t=5$$

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\log \frac{\sqrt{y}}{x^{4} \sqrt[3]{z}}$$

Estimate using the change of base formula. $$\log _{2} 20$$

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

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

If \(P\) dollars is deposited in a savings account that pays interest at a rate of \(r \%\) per year compounded continuously, find the balance after \(t\) years. $$P=100, \quad r=6 \frac{1}{2}, \quad t=10$$

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\ln \sqrt[4]{\frac{x^{7}}{y^{5} z}}$$

Estimate using the change of base formula. $$\log _{9} 0.2$$

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

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

Estimate using the change of base formula. $$\log _{6} \frac{1}{2}$$

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\ln x \sqrt[3]{\frac{y^{4}}{z^{3}}}$$

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

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

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. \(9-16:\) Write the expression as one logarithm. (a) \(\log _{3} x+\log _{3}(5 y) \quad\) (b) \(\log _{3}(2 z)-\log _{3} x\) (c) \(5 \log _{3} y\)

Evaluate using the change of base formula (without a calculator). $$\frac{\log _{5} 16}{\log _{5} 4}$$

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

An investment of \(P\) dollars increased to \(A\) dollars in \(t\) years. If interest was compounded continuously, find the interest rate. $$A=13,464, \quad P=1000, \quad t=20$$

Exer. \(9-16:\) Write the expression as one logarithm. (a) \(\log _{4}(3 z)+\log _{4} x \quad\) (b) \(\log _{4} x-\log _{4}(7 y)\) (c) \(\frac{1}{3} \log _{4} w\)

Evaluate using the change of base formula (without a calculator). $$\frac{\log _{7} 243}{\log _{7} 3}$$

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

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

An investment of \(P\) dollars increased to \(A\) dollars in \(t\) years. If interest was compounded continuously, find the interest rate. $$A=890.20, \quad P=400, \quad t=16$$

Change to logarithmic form. (a) \(10^{5}=100,000\) (b) \(10^{-3}=0.001\) (c) \(10^{x}=y+1\) (d) \(e^{T}=P\) (e) \(e^{2 t}=3-x\)

Sketch the graph of \(f\) if \(a=2\) (a) \(f(x)=a^{x}\) (b) \(f(x)=-a^{x}\) (c) \(f(x)=3 a^{x}\) (d) \(f(x)=a^{x+3}\) (e) \(f(x)=a^{x}+3\) (f) \(f(x)=a^{x-3}\) (g) \(f(x)=a^{x}-3\) (h) \(f(x)=a^{-x}\) (i) \(f(x)=\left(\frac{1}{a}\right)^{x}\) (j) \(f(x)=a^{3-x}\)

Exer. \(9-16:\) Write the expression as one logarithm. $$2 \log _{x} x+\frac{1}{3} \log _{2}(x-2)-5 \log _{x}(2 x+3)$$

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