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. $$\ln x=2$$

Evaluate each expression without using a calculator. $$\log 10^{7}$$

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

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\) the anmual 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. $$\ln x=3$$

Evaluate each expression without using a calculator. $$\log 10^{8}$$

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

The formula \(S=C(1+r)^{t}\) models inflation, where \(C=\) the value today, \(r=\) the anmual 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. $$\ln x=-3$$

Evaluate each expression without using a calculator. $$10^{\log 33}$$

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

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. $$\ln x=-4$$

Evaluate each expression without using a calculator. $$10^{\log 53}$$

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

Use a calculator with an \(\overline{e^{x}}\). key to solve. As of July \(2010,500\) million people worldwide shared versions of their lives on Facebook. The graph shows the number of active Facebook users (users who returned to the site within 30 days) for selected months from 2009 through 2010. (BAR GRAPH CAN'T COPY) The data can be modeled by $$f(x)=19 x+127 \quad \text { and } \quad g(x)=152.6 e^{0.0667 x}$$ in which \(f(x)\) and \(g(x)\) represent the number of active Facebook users, in millions, \(x\) months after December 2008 . a. According to the linear model, how many millions of active Facebook users were there in July 2010 , 19 months after December \(2008 ?\) b. According to the exponential model, how many millions of active Facebook users were there in July \(2010 ?\) c. Which function is a better model for the data in July \(2010 ?\)

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

Evaluate each expression without using a calculator. $$\ln 1$$

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

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

Evaluate each expression without using a calculator. $$\ln e$$

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

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

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

Evaluate each expression without using a calculator. $$\ln e^{6}$$

The function$$f(x)=\frac{90}{1+270 e^{-0.122 x}}$$ models the percentage, \(f(x)\), of people \(x\) years old with some coronary heart disease. Use this function to solve exercise. Round answers to the nearest tenth of a percent. Evaluate \(f(30)\) and describe what this means in practical terms.

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

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

Evaluate each expression without using a calculator. $$\ln e^{7}$$

The function$$f(x)=\frac{90}{1+270 e^{-0.122 x}}$$ models the percentage, \(f(x)\), of people \(x\) years old with some coronary heart disease. Use this function to solve exercise. Round answers to the nearest tenth of a percent. Evaluate \(f(70)\) and describe what this means in practical terms.

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

Evaluate each expression without using a calculator. $$\ln \frac{1}{e^{6}}$$

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 indicates that there is a limit to the epidemic's growth. (GRAPH CAN'T COPY) 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?

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

Evaluate each expression without using a calculator. $$\ln \frac{1}{e^{7}}$$

What is an exponential function?

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

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$\log _{0.1} 17$$

Evaluate each expression without using a calculator. $$e^{\ln 125}$$

What is the natural exponential function?

Evaluate each expression without using a calculator. $$e^{\ln 300}$$

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

Use a calculator to obtain an approximate value for \(e\) to as many decimal places as the display permits. Then use the 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.

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

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

Simplify each expression. $$\ln e^{9 x}$$

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. $$\log _{2}(x-1)+\log _{2}(x+1)=3$$