Chapter 5

Sketch a graph of \(f\) $$f(x)=\log _{4} 2 x$$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=2 x+1, \quad g(x)=4 x^{3}-5 x^{2} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$4\left(3^{x}\right)-3=13$$

\(\$ 2000\) at \(10 \%\) compounded continuously for 8 years

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$\log _{2} 25$$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=\sqrt{x-1}, \quad g(x)=3 x $$

Restrict the domain of \(f(x)\) so that \(f\) is one to-one. Then find \(f^{-1}(x)\). Answers may vary. $$ f(x)=x^{2 / 3}+1 $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$5\left(7^{x}\right)+3=83$$

\(\$ 100\) at \(19 \%\) compounded continuously for 50 years

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \log _{3} 67 $$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=\frac{x-3}{2}, \quad g(x)=2 x+3 $$

Restrict the domain of \(f(x)\) so that \(f\) is one to-one. Then find \(f^{-1}(x)\). Answers may vary. $$ f(x)=2(x+3)^{2 / 3} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$e^{x}+1=24$$

\(\$ 1600\) at \(10.4 \%\) compounded monthly for 2.5 years

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \log _{5} 130 $$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=1-5 x, \quad g(x)=\frac{1-x}{5} $$

Restrict the domain of \(f(x)\) so that \(f\) is one to-one. Then find \(f^{-1}(x)\). Answers may vary. $$ f(x)=\sqrt{9-2 x^{2}} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$1-2 e^{x}=-5$$

\(\$ 2000\) at \(8.7 \%\) compounded annually for 5 years

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \log _{6} 0.77 $$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=\sqrt[3]{x-1}, \quad g(x)=x^{3}+1 $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$2^{x}+1=15$$

Investments Compare investing \(\$ 2000\) at \(10 \%\) compounded monthly for 20 years with investing \(\$ 2000\) at \(13 \%\) compounded monthly for 20 years.

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \log _{2} 5+\log _{2} 7 $$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=\frac{1}{k x}, k>0, \quad g(x)=\frac{1}{k x}, k>0 $$

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=5 x-15 $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$3 \cdot 5^{x}=125$$

Lake Property In some states, lake shore property is increasing in value by \(15 \%\) per year. Determine the value of a \(\$ 90,000\) lake lot after 5 years.

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \log _{9} 85+\log _{7} 17 $$

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=a x^{2}, a>0, \quad g(x)=\sqrt{a x}, a>0 $$

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

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$5 e^{x}+2=20$$

College Tuition If college tuition is currently \(\$ 8000\) per year, inflating at \(6 \%\) per year, what will be the cost of tuition in 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. $$ f(x)=\sqrt[3]{x-5} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$6-2 e^{3 x}=-10$$

Doubling Time How long does it take for an investment to double its value if the interest is \(12 \%\) compounded annually? 6\% compounded annually?

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ 2 \log _{5} 15+\sqrt[3]{\log _{3} 67} $$

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=6-7 x $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$8-3(2)^{0.5 x}=-40$$

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \frac{\log _{2} 12}{\log _{2} 3} $$

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=\frac{x-5}{4} $$

Solve each equation. Use the change of base formula to approximate exact answers to the nearest hundredth when appropriate. $$2(3)^{-2 x}+5=167$$

Population Growth The population of Phoenix, Arizona, was 1.3 million in 2000 and growing continuously at a \(3 \%\) rate. (a) Assuming this trend continues, estimate the population of Phoenix in 2010 . (b) Determine graphically or numerically when this population might reach 2 million.

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \frac{\log _{7} 125}{\log _{7} 25} $$

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=\frac{x+2}{9} $$

Solve each equation. Approximate answers to four decimal places when appropriate. (a) \(\log x=2\) (b) \(\log x=-3\) (c) \(\log x=1.2\)

Federal Debt In fiscal year 2008 the federal budget deficit was about \(\$ 340\) billion. At the same time, 30 -year treasury bonds were paying \(4.54 \%\) interest. Suppose the American taxpayer loaned \(\$ 340\) billion to the federal government at \(4.54 \%\) compounded annually. If the federal government waited 30 years to pay the entire amount back, including the interest, how much would this be?

Solve the equation graphically. Express any solutions to the nearest thousandth. $$ \log _{2}\left(x^{3}+x^{2}+1\right)=7 $$

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

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