Chapter 5

Solve each equation. Approximate answers to four decimal places when appropriate. (a) \(\log x=1\) (b) \(\log x=-4\) (c) \(\log x=0.3\)

Solve the equation graphically. Express any solutions to the nearest thousandth. $$ \log _{3}\left(1+x^{2}+2 x^{4}\right)=4 $$

Exercises 77 and 78: Numerical representations for the functions \(f\) and \(g\) are given. Evaluate the expression, if possible. $$ \begin{array}{llll} \text { (a) }(g \circ f)(1) & \text { (b) }(f \circ g)(4) & \text { (c) }(f \circ f)(3) \end{array} $$ $$ \begin{array}{rrrrr} x & 1 & 3 & 4 & 6 \\ f(x) & 2 & 6 & 5 & 7 \end{array} $$ $$ \begin{array}{rrrrr} x & 2 & 3 & 5 & 7 \\ g(x) & 4 & 2 & 6 & 0 \end{array} $$

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=\sqrt{5-2 x}, x \leq \frac{5}{2} $$

Solve each equation. Approximate answers to four decimal places when appropriate. (a) \(\log _{2} x=6\) (b) \(\log _{3} x=-2\) (c) \(\ln x=2\)

Annuity If \(x\) dollars is deposited every 2 weeks \((26\) times per year) into an account paying an annual interest rate \(r,\) expressed in decimal form, then the amount \(A\) in the account after \(t\) years can be approximated by the formula $$ A=x\left(\frac{(1+r / 26)^{26 t}-1}{(r / 26)}\right) $$ If \(\$ 50\) is deposited every 2 weeks into an account paying \(8 \%\) interest, approximate the amount after 10 years.

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=\frac{1}{x+3} $$

Solve each equation. Approximate answers to four decimal places when appropriate. (a) \(\log _{4} x=2\) (b) \(\log _{8} x=-1\) (c) \(\ln x=-2\)

Solve the equation graphically. Express any solutions to the nearest thousandth. $$ \ln \left(x^{2}+2\right)=\log _{2}\left(10-x^{2}\right) $$

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=\frac{2}{x-1} $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$\log _{2} x=1.2$$

Continuous Compounding Over 5 years, the total value of a mutual fund account decreases continuously by \(15 \%\). Find a formula \(A(x)\) that calculates the amount of money in the account after \(x\) years.

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=2 x^{3} $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$\log _{4} x=3.7$$

Continuous Compounding A sum of money \(P\) in an account receives continuous interest and triples in 15 years. Find a formula \(A(x)\) that calculates the amount of money in the account after \(x\) years.

\(y=b x^{a}\) is used in applications involving biology and allometry. Another form of this equation is \(\log y=\log b+a \log x .\) Use properties of logarithms to obtain this second equation from the first. (Source: H. Lancaster, Quantitative Methods in Biological and Medical Sciences.)

Exercises \(81-94:\) (Refer to Example \(11 .\) ) Find functions \(f\) and \(g\) so that \(h(x)=(g \circ f)(x) .\) Answers may vary. $$ h(x)=(x+2)^{4} $$

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=1-4 x^{3} $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$5 \log _{7} 2 x=10$$

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=x^{2}, x \geq 0 $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$2 \log _{4} x=3.4$$

Exercises \(81-94:\) (Refer to Example \(11 .\) ) Find functions \(f\) and \(g\) so that \(h(x)=(g \circ f)(x) .\) Answers may vary. $$ h(x)=5(x+2)^{2}-4 $$

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=\sqrt[3]{1-x} $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$2 \log x=6$$

Light Absorption When sunlight passes through lake water, its initial intensity \(I_{0}\) decreases to a weaker intensity \(I\) at a depth of \(x\) feet according to the formula $$ \ln I-\ln I_{0}=-k x $$ where \(k\) is a positive constant. Solve this equation for \(I .\) (PICTURE NOT COPY)

Use the table for \(f(x)\) to find a table for \(\boldsymbol{f}^{-1}(\boldsymbol{x})\). Identify the domains and ranges of \(\boldsymbol{f}\) and \(\boldsymbol{f}^{-1}\) $$ \begin{array}{rrrr} x & 1 & 2 & 3 \\ f(x) & 5 & 7 & 9 \end{array} $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$\log 4 x=2$$

Solve each equation. Approximate answers to four decimal places when appropriate. $$2 \log 5 x=4$$

Population Growth The population \(P\) (in millions) of California \(x\) years after 2000 can be modeled by \(P=34 e^{0.013 x}\) A. Use properties of logarithms to solve this equation for \(x\) B. Use your equation to find \(x\) when \(P=38\). Interpret your answer.

Exercises \(81-94:\) (Refer to Example \(11 .\) ) Find functions \(f\) and \(g\) so that \(h(x)=(g \circ f)(x) .\) Answers may vary. $$ h(x)=\left(x^{3}-1\right)^{2} $$

Use the table for \(f(x)\) to find a table for \(\boldsymbol{f}^{-1}(\boldsymbol{x})\). Identify the domains and ranges of \(\boldsymbol{f}\) and \(\boldsymbol{f}^{-1}\) $$ \begin{array}{cccc} x & 0 & 2 & 4 \\ f(x) & 0 & 4 & 16 \end{array} $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$6-\log x=3$$

\(\quad\) The population \(P\) (in millions) of Georgia \(x\) years after 2000 can be modeled by \(P=8 e^{0.023 x}\) A. Use properties of logarithms to solve this equation for \(x\) B. Use your equation to find \(x\) when \(P=10\). Interpret your answer.

Use the table for \(f(x)\) to find a table for \(\boldsymbol{f}^{-1}(\boldsymbol{x})\). Identify the domains and ranges of \(\boldsymbol{f}\) and \(\boldsymbol{f}^{-1}\) $$ \begin{array}{cccc} x & 0 & 1 & 2 \\ f(x) & 1 & 2 & 4 \end{array} $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$4 \ln x=3$$

Solve \(A=P e^{n t}\) for \(t\)

Use \(f(x)\) to complete the table. $$ f(x)=4 x $$ TABLE CANNOT COPY.

Solve each equation. Approximate answers to four decimal places when appropriate. $$\ln 5 x=8$$

Solve \(P=P_{0} e^{r\left(t-t_{0}\right)}+5\) for \(t\)

Exercises \(81-94:\) (Refer to Example \(11 .\) ) Find functions \(f\) and \(g\) so that \(h(x)=(g \circ f)(x) .\) Answers may vary. $$ h(x)=5 \sqrt{x-1} $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$5 \ln x-1=6$$

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ f^{-1}(3) $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$2 \ln 3 x=8$$

Show that $$ \log _{2}(x+\sqrt{x^{2}-4})+\log _{2}(x-\sqrt{x^{2}-4})=2 $$ is an identity. What is the domain of the expression on the left side of the equation?

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ f^{-1}(5) $$

Solve each equation. Approximate answers to four decimal places when appropriate. $$4 \log _{2} x=16$$

A student insists that \(\log (x+y)\) and \(\log x+\log y\) are equal. How could you convince the student otherwise?

Exercises \(81-94:\) (Refer to Example \(11 .\) ) Find functions \(f\) and \(g\) so that \(h(x)=(g \circ f)(x) .\) Answers may vary. $$ h(x)=x^{3 / 4}-x^{1 / 4} $$

Use the tables to evaluate the following. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & 3 & 5 & 4 & 2 \end{array} $$ $$ \begin{array}{cccccc} x & -1 & 1 & 2 & 3 & 4 \\ g(x) & 0 & 2 & 1 & 4 & 5 \end{array} $$ $$ g^{-1}(4) $$

Radioactive Cesium-137 Radioactive cesium-137 was emitted in large amounts in the Chernobyl nuclear power station accident in Russia. The amount of a 100 - milligram sample of cesium remaining after \(x\) years can be described by \(A(x)=100 e^{-0.02295 x}\). (a) How much remains after 50 years? Is the half-life of cesium more or less than 50 years? (b) Estimate graphically the half-life of cesium- 137 .