Chapter 4

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\frac{1}{3}\left(\log _{4} x-\log _{4} y\right)+2 \log _{4}(x+1)\)

\(\text {Use a calculator with a } \overline{y^{x}} | \text { key or } a \ \bar{\wedge} \text { key to solve}.\). 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

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

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\frac{1}{3}\left[2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right]\)

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. CAN'T COPY THE GRAPH $$h(x)=\ln (2 x)$$

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. Round answers to the nearest dollar. If the inflation rate is \(6 \%,\) how much will a house now worth \(\$ 465,000\) be worth in 10 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. $$\log _{5} x+\log _{5}(4 x-1)=1$$

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. CAN'T COPY THE GRAPH $$h(x)=\ln \left(\frac{1}{2} x\right)$$

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(\frac{1}{3}\left[5 \ln (x+6)-\ln x-\ln \left(x^{2}-25\right)\right]\)

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. Round answers to the nearest dollar. If the inflation rate is \(3 \%,\) how much will a house now worth \(\$ 510,000\) be worth in 5 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. $$\log _{6}(x+5)+\log _{6} x=2$$

Use properties of logarithms to condense 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. $$\log _{3}(x+6)+\log _{3}(x+4)=1$$

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. CAN'T COPY THE GRAPH $$g(x)=\frac{1}{2} \ln x$$

Use properties of logarithms to condense 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 common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{5} 13\)

\(\text {Use a calculator with an }\left[e^{x}\right] \text { key to solve}\). Average annual premiums for employer-sponsored family health insurance policies more than doubled over 11 years. The bar graph shows the average cost of a family health insurance plan in the United States for six selected years from 2000 through 2011 . (BAR GRAPH CAN'T COPY) The data can be modeled by $$f(x)=782 x+6564 \text { and } g(x)=6875 e^{0.077 x}$$ in which \(f(x)\) and \(g(x)\) represent the average cost of a family health insurance plan \(x\) years after \(2000 .\) Use these functions to solve.Where necessary, round answers to the nearest whole dollar. a. According to the linear model, what was the average cost of a family health insurance plan in \(2011 ?\) b. According to the exponential model, what was the average cost of a family health insurance plan in \(2011 ?\) c. Which function is a better model for the data in \(2011 ?\)

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

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. CAN'T COPY THE GRAPH $$h(x)=\ln (-x)$$

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{6} 17\)

\(\text {Use a calculator with an }\left[e^{x}\right] \text { key to solve}\). Average annual premiums for employer-sponsored family health insurance policies more than doubled over 11 years. The bar graph shows the average cost of a family health insurance plan in the United States for six selected years from 2000 through 2011 . (BAR GRAPH CAN'T COPY) The data can be modeled by $$f(x)=782 x+6564 \text { and } g(x)=6875 e^{0.077 x}$$ in which \(f(x)\) and \(g(x)\) represent the average cost of a family health insurance plan \(x\) years after \(2000 .\) Use these functions to solve.Where necessary, round answers to the nearest whole dollar. a. According to the linear model, what was the average cost of a family health insurance plan in \(2008 ?\) b. According to the exponential model, what was the average cost of a family health insurance plan in \(2008 ?\) c. Which function is a better model for the data in \(2008 ?\)

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

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. CAN'T COPY THE GRAPH $$g(x)=2-\ln x$$

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{14} 87.5\)

\(\text {Use a calculator with an }\left[e^{x}\right] \text { key to solve}\). Average annual premiums for employer-sponsored family health insurance policies more than doubled over 11 years. The bar graph shows the average cost of a family health insurance plan in the United States for six selected years from 2000 through 2011 . (BAR GRAPH CAN'T COPY) The data can be modeled by $$f(x)=782 x+6564 \text { and } g(x)=6875 e^{0.077 x}$$ in which \(f(x)\) and \(g(x)\) represent the average cost of a family health insurance plan \(x\) years after \(2000 .\) Use these functions to solve.Where necessary, round answers to the nearest whole dollar. In college, we study large volumes of information information 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).

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. $$2 \log _{3}(x+4)=\log _{3} 9+2$$

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. CAN'T COPY THE GRAPH $$g(x)=1-\ln x$$

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{16} 57.2\)

\(\text {Use a calculator with an }\left[e^{x}\right] \text { key to solve}\). Average annual premiums for employer-sponsored family health insurance policies more than doubled over 11 years. The bar graph shows the average cost of a family health insurance plan in the United States for six selected years from 2000 through 2011 . (BAR GRAPH CAN'T COPY) The data can be modeled by $$f(x)=782 x+6564 \text { and } g(x)=6875 e^{0.077 x}$$ in which \(f(x)\) and \(g(x)\) represent the average cost of a family health insurance plan \(x\) years after \(2000 .\) Use these functions to solve.Where necessary, round answers to the nearest whole dollar. 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 have been worth in the year 2010 if interest were compounded a. monthly? b. 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. $$3 \log _{2}(x-1)=5-\log _{2} 4$$

Make Sense? In Exercises \(73-76\), determine whether each statement makes sense or does not make sense, and explain your reasoning. When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.

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

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{0,1} 17\)

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. $$\log _{2}(x-6)+\log _{2}(x-4)-\log _{2} x=2$$

Make Sense? In Exercises \(73-76\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Because carbon-14 decays exponentially, carbon dating can determine the ages of ancient fossils.

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

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{0.3} 19\)

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

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

What is an exponential function?

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{\pi} 63\)

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. $$\log (x+4)=\log x+\log 4$$

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

What is the natural exponential function?

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. \(\log _{\pi} 400\)

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. $$\log (5 x+1)=\log (2 x+3)+\log 2$$

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

Use a calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for \(x=10,100,1000\) \(10,000,100,000,\) and \(1,000,000 .\) Describe what happens to the expression as \(x\) increases

Use a graphing utility and the change-of-base property to graph each function. \(y=\log _{3} x\)