Chapter 4

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{5} 5 $$

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ g(x)=e^{x-1} $$

Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A=112.5 e^{0.012 t}\) describes Mexico's population, \(A,\) in millions, \(t\) years after 2010 . a. What is Mexico's growth rate? b. How long will it take Mexico to double its population?

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{2} \sqrt[5]{\frac{x y^{4}}{16}} $$

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. $$ e^{4 x-5}-7=11,243 $$

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{11} 11 $$

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ g(x)=e^{x+1} $$

The logistic growth function $$ f(t)=\frac{100,000}{1+5000 e^{-t}} $$ describes the number of people, \(f(t),\) who have become ill with influenza \(t\) weeks after its initial outbreak in a particular community. a. How many people became ill with the flu when the epidemic began? b. How many people were ill by the end of the fourth week? c. What is the limiting size of the population that becomes ill?

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \left[\frac{x^{3} \sqrt{x^{2}+1}}{(x+1)^{4}}\right] $$

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. $$ 7^{x+2}=410 $$

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{4} 1 $$

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ g(x)=e^{x}+2 $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { World Population }} \\ {\text { after } 1949} & {\text { (billions) }} \\ {1(1950)} & {2.6} \\ {11(1960)} & {3.0} \\ {21(1970)} & {3.7} \\ {21(1970)} & {4.5} \\ {41(1990)} & {5.3} \\ {51(2000)} & {6.1} \\ {61(2010)} & {6.9} \end{array} $$ We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) How well does the function model the data showing a world population of 6.1 billion for \(2000 ?\)

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \left[\frac{x^{4} \sqrt{x^{2}+3}}{(x+3)^{5}}\right] $$

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. $$ 5^{x-3}=137 $$

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{6} 1 $$

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ g(x)=e^{x}-1 $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { World Population }} \\ {\text { after } 1949} & {\text { (billions) }} \\ {1(1950)} & {2.6} \\ {11(1960)} & {3.0} \\ {21(1970)} & {3.7} \\ {21(1970)} & {4.5} \\ {41(1990)} & {5.3} \\ {51(2000)} & {6.1} \\ {61(2010)} & {6.9} \end{array} $$ We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) How well does the function model the data showing a world population of 6.9 billion for \(2010 ?\)

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. $$ 7^{0.3 x}=813 $$

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{5} 5^{7} $$

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ h(x)=e^{x-1}+2 $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { World Population }} \\ {\text { after } 1949} & {\text { (billions) }} \\ {1(1950)} & {2.6} \\ {11(1960)} & {3.0} \\ {21(1970)} & {3.7} \\ {21(1970)} & {4.5} \\ {41(1990)} & {5.3} \\ {51(2000)} & {6.1} \\ {61(2010)} & {6.9} \end{array} $$ We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) When did world population reach 7 billion?

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left[\frac{100 x^{3} \sqrt[3]{5-x}}{3(x+7)^{2}}\right] $$

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. $$ 3^{\frac{x}{7}}=0.2 $$

In Exercises 21–42, evaluate each expression without using a calculator. $$ \log _{4} 4^{6} $$

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ h(x)=e^{x+1}-1 $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { World Population }} \\ {\text { after } 1949} & {\text { (billions) }} \\ {1(1950)} & {2.6} \\ {11(1960)} & {3.0} \\ {21(1970)} & {3.7} \\ {21(1970)} & {4.5} \\ {41(1990)} & {5.3} \\ {51(2000)} & {6.1} \\ {61(2010)} & {6.9} \end{array} $$ We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) When will world population reach 8 billion?

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. $$ \log 5+\log 2 $$

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. $$ 5^{2 x+3}=3^{x-1} $$

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ h(x)=e^{-x} $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { World Population }} \\ {\text { after } 1949} & {\text { (billions) }} \\ {1(1950)} & {2.6} \\ {11(1960)} & {3.0} \\ {21(1970)} & {3.7} \\ {21(1970)} & {4.5} \\ {41(1990)} & {5.3} \\ {51(2000)} & {6.1} \\ {61(2010)} & {6.9} \end{array} $$ We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) According to the model, what is the limiting size of the population that Earth will eventually sustain?

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. $$ \log 250+\log 4 $$

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. $$ 7^{2 x+1}=3^{x+2} $$

In Exercises 21–42, evaluate each expression without using a calculator. $$ 7^{\log _{7} 23} $$

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ h(x)=-e^{x} $$

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) What percentage of 20 -year-olds have some coronary heart disease?

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. $$ \ln x+\ln 7 $$

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. $$ e^{2 x}-3 e^{x}+2=0 $$

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

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) What percentage of 80 -year-olds have some coronary heart disease?

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. $$ \ln x+\ln 3 $$

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. $$ e^{2 x}-2 e^{x}-3=0 $$

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

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ g(x)=\frac{1}{2} e^{x} $$

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) At what age is the percentage of some coronary heart disease \(50 \% ?\)

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

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. $$ e^{4 x}+5 e^{2 x}-24=0 $$

Graph \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ h(x)=e^{2 x}+1 $$

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) At what age is the percentage of some coronary heart disease \(70 \% ?\)