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

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

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

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

Explaining the Concepts Suppose that a population that is growing exponentially increases from \(800,000\) people in 2010 to \(1,000,000\) people in \(2013 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

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. $$ 4 \ln (x+6)-3 \ln x $$

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

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

Explaining the Concepts What is the half-life of a substance?

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. $$ 8 \ln (x+9)-4 \ln x $$

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

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

Explaining the Concepts Describe a difference between exponential growth and logistic growth.

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. $$ 3 \ln x+5 \ln y-6 \ln z $$

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. $$ 5 \ln (2 x)=20 $$

Explaining the Concepts Describe the shape of a scatter plot that suggests modeling the data with an exponential function.

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. $$ 4 \ln x+7 \ln y-3 \ln z $$

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. $$ 6 \ln (2 x)=30 $$

Explaining the Concepts You take up weightlifting and record the maximum number of pounds you can lift at the end of each week. You start off with rapid growth in terms of the weight you can lift from week to week, but then the growth begins to level off. Describe how to obtain a function that models the number of pounds you can lift at the end of each week. How can you use this function to predict what might happen if you continue the sport?

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. $$ \frac{1}{2}(\log x+\log y) $$

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. $$ 6+2 \ln x=5 $$

Explaining the Concepts Would you prefer that your salary be modeled exponentially or logarithmically? Explain your answer.

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. $$ \frac{1}{3}\left(\log _{4} x-\log _{4} y\right) $$

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. $$ 7+3 \ln x=6 $$

Explaining the Concepts One problem with all exponential growth models is that nothing can grow exponentially forever. Describe factors that might limit the size of a population.

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. $$ \frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1) $$

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. $$ \ln \sqrt{x+3}=1 $$

Use a calculator with a \(y^{x}\) key or a \(A\) 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?

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ a. Use your graphing utility's exponential regression option to obtain a model of the form \(y=a b^{x}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data? b. Rewrite the model in terms of base \(e .\) By what percentage is the population of the United States increasing each year?

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. $$ \frac{1}{3}\left(\log _{4} x-\log _{4} y\right)+2 \log _{4}(x+1) $$

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. $$ \ln \sqrt{x+4}=1 $$

The figure shows the graph of \(f(x)=\ln x .\) In Exercises \(65-74\) 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. $$ g(x)=\ln (x+1) $$

Use a calculator with a \(y^{x}\) key or a \(A\) 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 \(f(x)=1000(0.5)^{\frac{1}{30}}\) 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 .

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ Use your graphing utility's logarithmic regression option to obtain a model of the form \(y=a+b \ln x\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

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. $$ \frac{1}{3}\left[2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right] $$

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

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ Use your graphing utility's linear regression option to obtain a model of the form \(y=a x+b\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

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. $$ \frac{1}{3}\left[5 \ln (x+6)-\ln x-\ln \left(x^{2}-25\right)\right] $$

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. $$ \log _{6}(x+5)+\log _{6} x=2 $$

Use a calculator with a \(y^{x}\) key or a \(A\) key to solve. If the inflation rate is \(3 \%,\) how much will a house now worth \(\$ 510,000\) be worth in 5 years?

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ Use your graphing utility's power regression option to obtain a model of the form \(y=a x^{b}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

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. $$ \log x+\log \left(x^{2}-1\right)-\log 7-\log (x+1) $$

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. $$ \log _{3}(x+6)+\log _{3}(x+4)=1 $$

Use a calculator with a \(y^{x}\) key or a \(A\) key to solve. A decimal approximation for \(\sqrt{3}\) is 1.7320508 . Use a calculator to find \(2^{1.7}, 2^{1.73}, 2^{1.73}, 2^{1.73215},\) and \(2^{17300508} .\) Now find \(2^{\sqrt{3}}\). What do you observe?

In Example I on page \(520,\) we used two data points and an exponential function to model the population of the United States from 1970 through 2010 . The data are shown again in the table. Use all five data points to solve Exercises \(66-70\). $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { U.S. Population }} \\ {\text { after } 1969} & {\text { (millions) }} \\ {1(1970)} & {203.3} \\ {11(1980)} & {226.5} \\ {21(1990)} & {248.7} \\ {31(2000)} & {281.4} \\ {41(2010)} & {308.7} \end{array} $$ Use the values of \(r\) in Exercises \(66-69\) to select the two model= of best fit. Use each of these models to predict by which yeathe U.S. population will reach 335 million. How do these answers compare to the year we found in Example \(1,\) namel \(=\) \(2020 ?\) If you obtained different years, how do you account fo this difference?

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. $$ \log x+\log \left(x^{2}-4\right)-\log 15-\log (x+2) $$

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. $$ \log _{6}(x+3)+\log _{6}(x+4)=1 $$

Use a calculator with a \(y^{x}\) key or a \(A\) key to solve. A decimal approximation for \(\sqrt{3}\) is 1.7320508 . Use a calculator to find \(2^{1.7}, 2^{1.73}, 2^{1.73}, 2^{1.73215},\) and \(2^{17300508} .\) Now find \(2^{\sqrt{3}}\). What do you observe?

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{5} 13 $$

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. $$ \log _{2}(x+2)-\log _{2}(x-5)=3 $$