Chapter 4

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{3} 405-\log _{3} 5$$ Condensing the common logarithmic expression, using properties of loga function, quotient rule and power rule of logarithms, we have sh that

Solve each exponential equation . Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ e^{4 x}-3 e^{2 x}-18=0 $$

Graph \(f(x)=\left(\frac{1}{4}\right)^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (2 x+5)-\log x $$

Solve each exponential equation . Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 3^{2 x}+3^{x}-2=0 $$

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=3^{x} \text { and } g(x)=3^{-x} $$

Exercises \(47-52\) present data in the form of tables. For each data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Exercise \(72,\) you will use your graphing utility to obtain these functions.) Savings Needed for Health-Care Expenses during Retirement $$ \begin{array}{cc} {\text { Age at Death }} & {\text { Savings Needed }} \\ {80} & {\$ 219,000} \\ {85} & {\$ 307,000} \\ {90} & {\$ 409,000} \\ {95} & {\$ 524,000} \\ {100} & {\$ 656,000} \end{array} $$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (3 x+7)-\log x $$

Solve each exponential equation . Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 2^{2 x}+2^{x}-12=0 $$

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=3^{x} \text { and } g(x)=-3^{x} $$

Exercises \(47-52\) present data in the form of tables. For each data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Exercise \(72,\) you will use your graphing utility to obtain these functions.) Intensity and Loudness Level of Various Sounds $$\begin{array}{lc}{\underline{\phantom{xx}}} & {\text { Loudness Level }} \\ \text { Intensity (watts per meter') } & {\text { (decibels) }} \\ {0.1 \text { (loud thunder) }} & {110} \\ {1 \text { (rock concert, } 2 \text { yd from speakers) }} & {120} \\ {10 \text { (jackhammer) }} & {130} \\ {100 \text { (jet takeoff, 40 yd away) }} & {140}\end{array}$$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log x+3 \log y $$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{3} x=4 $$

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=3^{x} \text { and } g(x)=\frac{1}{3} \cdot 3^{x} $$

Exercises \(47-52\) present data in the form of tables. For each data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Exercise \(72,\) you will use your graphing utility to obtain these functions.) Temperature Increase in an Enclosed Vehicle $$ \begin{array}{cc} {\text { Minutes }} & {\text { Temperature Increase }\left(^{\circ} \mathrm{F}\right)} \\ {10} & {19^{\circ}} \\ {20} & {29^{\circ}} \\ {30} & {34^{\circ}} \\ {40} & {38^{\circ}} \\ {50} & {41^{\circ}} \\ {60} & {43^{\circ}} \end{array} $$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log x+7 \log y $$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{5} x=3 $$

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=3^{x} \text { and } g(x)=3 \cdot 3^{x} $$

Exercises \(47-52\) present data in the form of tables. For each data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Exercise \(72,\) you will use your graphing utility to obtain these functions.) Hamachiphobia $$ \begin{array}{ccc} {} & {\text { Percentage }} & {\text { Percentage } W h o} \\ {} & {\text { Who Won't }} & {\text { Don't Approve of }} \\ {\text { Generation }} & {\text { Try Sushi }} & {\text { Marriage Equality }} \\\ {\text { Millennials }} & {42} & {36} \\ {\text { Gen } X} & {52} & {36} \\ {\text { Boomers }} & {60} & {49} \\ {\text { Silent/Greatest }} & {72} & {66} \\ {\text { Generation }} \end{array} $$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{2} \ln x+\ln y $$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \ln x=2 $$

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=\left(\frac{1}{2}\right)^{x} \text { and } g(x)=\left(\frac{1}{2}\right)^{x-1}+1 $$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{3} \ln x+\ln y $$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \ln x=3 $$

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=\left(\frac{1}{2}\right)^{x} \text { and } g(x)=\left(\frac{1}{2}\right)^{x-1}+2 $$

Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=100(4.6)^{x} $$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 2 \log _{b} x+3 \log _{b} y $$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{4}(x+5)=3 $$

In Exercises \(53-58,\) begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$ g(x)=\log _{2}(x+1) $$

Round answers to the nearest cent. Find the accumulated value of an investment of \(\$ 10,000\) for 5 years at an interest rate of \(5.5 \%\) if the money is a. compounded semiannually; b. compounded quarterly: c. compounded monthly; d. compounded continuously.

Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=1000(7.3)^{x} $$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 5 \log _{b} x+6 \log _{b} y $$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{5}(x-7)=2 $$

In Exercises \(53-58,\) begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$ g(x)=\log _{2}(x+2) $$

Round answers to the nearest cent. Find the accumulated value of an investment of \(\$ 5000\) for 10 years at an interest rate of \(6.5 \%\) if the money is a. compounded semiannually, b. compounded quarterly; c. compounded monthly; d. compounded continuously.

Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=2.5(0.7)^{x} $$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 5 \ln x-2 \ln y $$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{2}(x+25)=4 $$

Round answers to the nearest cent. Suppose that you have \(\$ 12,000\) to invest. Which investment yields the greater return over 3 years: \(7 \%\) compounded monthly or \(6.85 \%\) compounded continuously?

Rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=4.5(0.6)^{x} $$

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 7 \ln x-3 \ln y $$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{2}(x+50)=5 $$

In Exercises \(53-58,\) begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$ h(x)=2+\log _{2} x $$

Round answers to the nearest cent. Suppose that you have \(\$ 6000\) to invest. Which investment yields the greater return over 4 years: \(8.25 \%\) compounded quarterly or \(8.3 \%\) compounded semiannually?

Explaining the Concepts Nigeria has a growth rate of 0.025 or \(2.5 \% .\) Describe what this means.

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 3 \ln x-\frac{1}{3} \ln y $$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{3}(x+4)=-3 $$

In Exercises \(53-58,\) begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$ g(x)=\frac{1}{2} \log _{2} x $$

graph f and g in the same rectangular coordinate system. Then find the point of intersection of the two graphs. $$ f(x)=2^{x}, g(x)=2^{-x} $$

Explaining the Concepts How can you tell whether an exponential model describes exponential growth or exponential decay?