Chapter 12

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. $$\log _{3} x=4$$

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. $$\log _{2} 96-\log _{2} 3$$

Evaluate each expression without using a calculator. $$8^{\log _{8} 19}$$

Use the compound interest formulas, \(A=P\left(1+\frac{r}{n}\right)^{n-1}\) and \(A=P e^{n t},\) to solve Exercises \(39-42 .\) 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. 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. $$\log _{5} x=3$$

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. $$\log _{3} 405-\log _{3} 5$$

Evaluate each expression without using a calculator. $$7^{\log _{7} 23}$$

Use the compound interest formulas, \(A=P\left(1+\frac{r}{n}\right)^{n-1}\) and \(A=P e^{n t},\) to solve Exercises \(39-42 .\) 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. 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. $$\log _{2} x=-4$$

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. $$\log (2 x+5)-\log x$$

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

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. $$\log _{2} x=-5$$

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. $$\log (3 x+7)-\log x$$

Graph \(f(x)=5^{x}\) and \(g(x)=\log _{5} x\) in the same rectangular coordinate system.

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. $$\log _{9} x=\frac{1}{2}$$

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. $$\log x+3 \log y$$

Graph \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\log _{\frac{1}{2}} x\) in the same rectangular coordinate system.

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. $$\log _{25} x=\frac{1}{2}$$

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. $$\log x+7 \log y$$

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

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. $$\log x=2$$

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. $$\frac{1}{2} \ln x+\ln y$$

Find the domain of each logarithmic function. $$f(x)=\log _{5}(x+4)$$

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. $$\log x=3$$

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. $$\frac{1}{3} \ln x+\ln y$$

Find the domain of each logarithmic function. $$f(x)=\log _{5}(x+6)$$

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. $$\log _{4}(x+5)=3$$

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. $$2 \log _{b} x+3 \log _{b} y$$

Find the domain of each logarithmic function. $$f(x)=\log (2-x)$$

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

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. $$\log _{5}(x-7)=2$$

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. $$5 \log _{b} x+6 \log _{b} y$$

Find the domain of each logarithmic function. $$f(x)=\log (7-x)$$

Graph \(f\) and \(g\) in the same rectangular coordinate system. Then find the point of intersection of the two graphs. $$f(x)=2^{x+1}, g(x)=2^{-x+1}$$

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. $$\log _{3}(x-4)=-3$$

Find the domain of each logarithmic function. $$f(x)=\ln (x-2)^{2}$$

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. $$5 \ln x-2 \ln y$$

Graph \(y=2^{x}\) and \(x=2^{y}\) in the same rectangular coordinate system.

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. $$\log _{7}(x+2)=-2$$

Find the domain of each logarithmic function. $$f(x)=\ln (x-7)^{2}$$

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. $$7 \ln x-3 \ln y$$

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

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. $$\log _{4}(3 x+2)=3$$

Evaluate each expression without using a calculator. $$\log 100$$

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. $$3 \ln x-\frac{1}{3} \ln y$$

Use a calculator with a \(\overline{y^{x}}\) key or \(a \ \triangle\) key to solve. India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential function \(f(x)=574(1.026)^{x}\) models the population of India, \(f(x),\) in millions, \(x\) years after 1974 a. Substitute 0 for \(x\) and, without using a calculator, find India's population in 1974 b. Substitute 27 for \(x\) and use your calculator to find India's population, to the nearest million, in the year 2001 as modeled by this function. c. Find India's population, to the nearest million, in the year 2028 as predicted by this function. d. Find India's population, to the nearest million, in the year 2055 as predicted by this function. e. What appears to be happening to India's population every 27 years?

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. $$\log _{2}(4 x+1)=5$$

Evaluate each expression without using a calculator. $$\log 1000$$

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. $$2 \ln x-\frac{1}{2} \ln y$$

Use a calculator with a \(\overline{y^{x}}\) key or \(a \ \triangle\) key to solve. The 1986 explosion at the Chernobyl nuclear power plant in the former Soviet Union sent about 1000 kilograms of radioactive cesium-137 into the atmosphere. The function \(f(x)=1000(0.5)^{\frac{x}{30}}\) describes the amount, \(f(x),\) in kilograms, of cesium-137 remaining in Chernobyl \(x\) years after \(1986 .\) If even 100 kilograms of cesium- 137 remain in Chernobyl's atmosphere, the area is considered unsafe for human habitation. Find \(f(80)\) and determine if Chernobyl will be safe for human habitation by 2066