Chapter 4

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log _{3} 405-\log _{3} 5 $$

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln x=3$$

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 \(f(x)=1000(0.5)^{x / 30}\) 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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log (2 x+5)-\log x $$

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$5 \ln (2 x)=20$$

It is 8: 00 P.M. and West Side Story is scheduled to begin. When the curtain does not go up, a rumor begins to spread through the 400-member audience: The lead roles of Tony and Maria might be understudied by Anthony Hopkins and Jodie Foster. The function $$f(x)=\frac{400}{1+399(0.67)^{x}}$$ models the number of people in the audience, \(f(x),\) who have heard the rumor \(x\) minutes after \(8: 00 .\) Use this function to solve. Evaluate \(f(10)\) and describe what this means in practical terms.

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log (3 x+7)-\log x $$

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$6 \ln (2 x)=30$$

It is 8: 00 P.M. and West Side Story is scheduled to begin. When the curtain does not go up, a rumor begins to spread through the 400-member audience: The lead roles of Tony and Maria might be understudied by Anthony Hopkins and Jodie Foster. The function $$f(x)=\frac{400}{1+399(0.67)^{x}}$$ models the number of people in the audience, \(f(x),\) who have heard the rumor \(x\) minutes after \(8: 00 .\) Use this function to solve. Evaluate \(f(20)\) and describe what this means in practical terms.

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log x+3 \log y $$

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$6+2 \ln x=5$$

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the graph's \(x\) -intercept? What is the vertical asymptote? $$g(x)=\log _{2}(x+1)$$

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\) the annual inflation rate, and \(S=\) the inflated value \(t\) years from now. Use this formula to solve, If the inflation rate is \(6 \%,\) how much will a house now worth \(\$ 65,000\) be worth in 10 years?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log x+7 \log y $$

In Exercises \(27-30,\) you worked with the logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ which models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use your graphing utility to graph the function in a \([0,100,10]\) by \([0,100,10]\) viewing rectangle. Describe as specifically as possible what the logistic curve indicates about aging and the percentage of Americans with coronary heart disease.

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$7+3 \ln x=6$$

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the graph's \(x\) -intercept? What is the vertical asymptote? $$g(x)=\log _{2}(x+2)$$

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\) the annual inflation rate, and \(S=\) the inflated value \(t\) years from now. Use this formula to solve, If the inflation rate is \(3 \%,\) how much will a house now worth \(\$ 110,000\) be worth in 5 years?

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln \sqrt{x+3}=1$$

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the graph's \(x\) -intercept? What is the vertical asymptote? $$h(x)=1+\log _{2} x$$

A decimal approximation for \(\sqrt{3}\) is \(1.7320508 .\) Use a calculator to find \(2^{1.7}, 2^{1.73}, 2^{1.732}, 2^{1.73205},\) and \(2^{1.7320508}\) Now find \(2^{\sqrt{3}} .\) What do you observe?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 2 \log _{4} x+3 \log _{b} y $$

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln \sqrt{x+4}=1$$

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the graph's \(x\) -intercept? What is the vertical asymptote? $$h(x)=2+\log _{2} x$$

A decimal approximation for \(\pi\) is \(3.141593 .\) Use a calculator to find \(2^{3}, 2^{3.1}, 2^{3.14}, 2^{3.141}, 2^{3.1415}, 2^{3.14159},\) and \(2^{3.141593} .\) Now find \(2^{\pi} .\) What do you observe?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 2 \log _{b} x+3 \log _{b} y $$

The World Health Organization makes predictions about the number of AIDS cases based on a compromise between a linear model and an exponential growth model. Explain why the World Health Organization does this.

Use the formula \(R=6 e^{12.77 x},\) where \(x\) is the blood alcohol concentration and \(R,\) given as a percent, is the risk of having a car accident, to solve Exercises \(53-54\) What blood alcohol concentration corresponds to a \(25 \%\) risk of a car accident?

The graph on the next page shows the number of Americans enrolled in HMOs, in millions, from 1992 through \(2000 .\) The data can be modeled by the exponential function $$f(x)=36.1 e^{0.113 x}$$ which describes enrollment in HMOs, \(f(x),\) in millions, \(x\) years after \(1992 .\) Use this function to solve. According to the model, how many Americans will be enrolled in HMOs in the year \(2006 ?\) Round to the nearest tenth of a million.

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the graph's \(x\) -intercept? What is the vertical asymptote? $$g(x)=\frac{1}{2} \log _{2} x$$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 5 \log _{6} x+6 \log _{b} y $$

Use the formula \(R=6 e^{12.77 x},\) where \(x\) is the blood alcohol concentration and \(R,\) given as a percent, is the risk of having a car accident, to solve Exercises \(53-54\) What blood alcohol concentration corresponds to a \(50 \%\) risk of a car accident?

Begin by graphing \(f(x)=\log _{2} x .\) Then use transformations of this graph to graph the given function. What is the graph's \(x\) -intercept? What is the vertical asymptote? $$g(x)=-2 \log _{2} x$$

The graph on the next page shows the number of Americans enrolled in HMOs, in millions, from 1992 through \(2000 .\) The data can be modeled by the exponential function $$f(x)=36.1 e^{0.113 x}$$ which describes enrollment in HMOs, \(f(x),\) in millions, \(x\) years after \(1992 .\) Use this function to solve. According to the model, how many Americans will be enrolled in HMOs in the year \(2008 ?\) Round to the nearest tenth of a million.

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 5 \ln x-2 \ln y $$

This activity is intended for three or four people who would like to take up weightlifting. Each person in the group should record the maximum number of pounds that he or she can lift at the end of each week for the first 10 consecutive weeks. Use the Logarithmic REGression option of a graphing utility to obtain a model showing the amount of weight that group members can lift from week 1 through week 10. Graph each of the models in the same viewing rectangle to observe similarities and differences among weight-growth patterns of each member. Use the functions to predict the amount of weight that group members will be able to lift in the future. If the group continues to work out together, check the accuracy of these predictions.

The formula \(A=18.9 e^{0.0055 t}\) models the population of New York State, \(A\), in millions, \(t\) years after 2000 . a. What was the population of New York in \(2000 ?\) b. When will the population of New York reach 19.6 million?

In college, we study large volumes of informationinformation that, unfortunately, we do not often retain for very long. The function $$f(x)=80 e^{-0.5 x}+20$$ describes the percentage of information, \(f(x),\) that a particular person remembers \(x\) weeks after learning the information. a. Substitute 0 for \(x\) and, without using a calculator, find the percentage of information remembered at the moment it is first learned. b. Substitute 1 for \(x\) and find the percentage of information that is remembered after 1 week. c. Find the percentage of information that is remembered after 4 weeks. d. Find the percentage of information that is remembered after one year ( 52 weeks).

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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 7 \ln x-3 \ln y $$

Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by exponential or logarithmic functions. Group members should select the two sets of data that are most interesting and relevant. For each data set selected, find a model that best fits the data. Each group member should make one prediction based on the model and then discuss a consequence of this prediction. What factors might change the accuracy of each prediction?

The formula \(A=15.9 e^{0.0235 t}\) models the population of Florida, \(A\), in millions, \(t\) years after 2000 . a. What was the population of Florida in \(2000 ?\) b. When will the population of Florida reach 17.5 million?

In \(1626,\) Peter Minuit convinced the Wappinger Indians to sell him Manhattan Island for \(\$ 24 .\) If the Native Americans had put the \(\$ 24\) into a bank account paying \(5 \%\) interest, how much would the investment be worth in the year 2000 if interest were compounded a. monthly? b. continuously?

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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 3 \ln x-\frac{1}{3} \ln y $$

The function $$N(t)=\frac{30,000}{1+20 e^{-1.5 t}}$$ describes the number of people, \(N(t),\) who become ill with influenza \(t\) weeks after its initial outbreak in a town with \(30,000\) inhabitants. The horizontal asymptote in the graph at the top of the next column indicates that there is a limit to the epidemic's growth. a. How many people became ill with the flu when the epidemic began? (When the epidemic began, \(t=0 .)\) b. How many people were ill by the end of the third week? c. Why can't the spread of an epidemic simply grow indefinitely? What does the horizontal asymptote shown in the graph indicate about the limiting size of the population that becomes ill?

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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 2 \ln x-\frac{1}{2} \ln y $$

What is an exponential function?

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