Chapter 4

Solve each exponential equation in Exercises \(23-48\). 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. $$ 3^{2 x}+3^{x}-2-0 $$

In Exercises \(41-70,\) 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 in Exercises \(23-48\). 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. $$ 2^{2 x}+2^{x}-12-0 $$

In Exercises \(41-70,\) 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 logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{3} x=4 $$

In Exercises \(41-70,\) 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. $$ \log x+3 \log y $$

In Exercises \(47-52,\) 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. $$ f(x)-3^{x} \text { and } g(x)-\frac{1}{3} \cdot 3^{x} $$

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{5} x=3 $$

In Exercises \(41-70,\) 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 in Exercises \(49-92 .\) 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. $$ \ln x=2 $$

In Exercises \(41-70,\) 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 in Exercises \(49-92 .\) 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. $$ \ln x=3 $$

In Exercises \(41-70,\) 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 in Exercises \(49-92 .\) 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. $$ \log _{4}(x+5)=3 $$

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. $$ g(x)=\log _{2}(x+1) $$

In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=100(4.6)^{x}$$

In Exercises \(41-70,\) 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. $$ 2 \log _{b} x+3 \log _{b} y $$

Use the compound interest formulas \(A-P\left(1+\frac{r}{n}\right)^{n t}\) and \(A-P e^{n}\) to solve Exercises \(53-56 .\) 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.

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{5}(x-7)=2 $$

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. $$ g(x)=\log _{2}(x+2) $$

In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=1000(7.3)^{x}$$

In Exercises \(41-70,\) 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. $$ 5 \log _{b} x+6 \log _{b} y $$

Use the compound interest formulas \(A-P\left(1+\frac{r}{n}\right)^{n t}\) and \(A-P e^{n}\) to solve Exercises \(53-56 .\) 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.

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{2}(x+25)=4 $$

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. $$ h(x)=1+\log _{2} x $$

In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=2.5(0.7)^{x}$$

In Exercises \(41-70,\) 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. $$ 5 \ln x-2 \ln y $$

Use the compound interest formulas \(A-P\left(1+\frac{r}{n}\right)^{n t}\) and \(A-P e^{n}\) to solve Exercises \(53-56 .\) 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?

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{2}(x+50)=5 $$

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. $$ h(x)=2+\log _{2} x $$

In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=4.5(0.6)^{x}$$

In Exercises \(41-70,\) 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. $$ 3 \ln x-\frac{1}{3} \ln y $$

Use the compound interest formulas \(A-P\left(1+\frac{r}{n}\right)^{n t}\) and \(A-P e^{n}\) to solve Exercises \(53-56 .\) 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?

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{3}(x+4)=-3 $$

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. $$ g(x)=\frac{1}{2} \log _{2} x $$

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

In Exercises \(41-70,\) 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 in Exercises \(49-92 .\) 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. $$ \log _{7}(x+2)=-2 $$

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. $$ g(x)=-2 \log _{2} x $$

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

In Exercises \(41-70,\) 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. $$ 2 \ln x-\frac{1}{2} \ln y $$

The figure shows the graph of \(f(x)=\log x .\) Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ g(x)=\log (x-1) $$

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{4}(3 x+2)=3 $$

Suppose that a population that is growing exponentially increases from \(800,000\) people in 2007 to \(1,000,000\) people in \(2010 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

In Exercises \(41-70,\) 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. $$ 4 \ln (x+6)-3 \ln x $$

In Exercises \(57-58,\) graph \(f\) and \(g\) in the same rectangular coordinate system. Then find the point of intersection of the two graphs. Graph \(y-2^{x}\) and \(x-2^{y}\) in the same rectangular coordinate system.

Solve each logarithmic equation in Exercises \(49-92 .\) 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. $$ \log _{2}(4 x+1)=5 $$

In Exercises \(41-70,\) 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. $$ 8 \ln (x+9)-4 \ln x $$

Graph \(y-3^{x}\) and \(x-3^{y}\) in the same rectangular coordinate system.

The figure shows the graph of \(f(x)=\log x .\) Use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. (GRAPH CANNOT COPY). $$ h(x)=\log x-1 $$