Chapter 5

The magnitude of a star is defined by the equation $$M=6-2.5 \log \frac{I}{I_{0}}$$ where \(I_{0}\) is the measure of a just-visible star and \(I\) is the actual intensity of the star being measured. The dimmest stars are of magnitude \(6,\) and the brightest are of magnitude 1. Determine the ratio of light intensities between a star of magnitude 1 and a star of magnitude 3.

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log (-x)$$

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$\left(\frac{1}{2}\right)^{\sqrt{2}}$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$\left(\frac{1}{2}\right)^{x}=5$$

For each statement, write an equivalent statement in exponential form. $$\log _{6} 36=2$$

Newton's law of cooling says that the rate at which an object cools is proportional to the difference \(C\) in temperature between the object and the environment around it. The temperature \(f(t)\) of the object at time t in appropriate units after being introduced into an environment with a constant temperature \(T_{0}\) is $$f(t)=T_{0}+C e^{-k t}$$ where \(C\) and \(k\) are constants. Use this result. Boiling water at \(100^{\circ} \mathrm{C}\) is placed in a freezer at \(0^{\circ} \mathrm{C}\). The temperature of the water is \(50^{\circ} \mathrm{C}\) after 24 minutes. Approximate the temperature of the water after 96 minutes.

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log \left(-\frac{1}{2} x\right)$$

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$\left(\frac{1}{3}\right)^{\sqrt{6}}$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$\left(\frac{1}{3}\right)^{x}=6$$

For each statement, write an equivalent statement in exponential form. $$\log _{5} 5=1$$

Newton's law of cooling says that the rate at which an object cools is proportional to the difference \(C\) in temperature between the object and the environment around it. The temperature \(f(t)\) of the object at time t in appropriate units after being introduced into an environment with a constant temperature \(T_{0}\) is $$f(t)=T_{0}+C e^{-k t}$$ where \(C\) and \(k\) are constants. Use this result. A pot of coffee with a temperature of \(100^{\circ} \mathrm{C}\) is set down in a room with a temperature of \(20^{\circ} \mathrm{C}\). The coffee cools to \(60^{\circ} \mathrm{C}\) after 1 hour. (a) Write an equation to model the data. (b) Estimate the temperature after a half hour. (c) About how long will it take for the coffee to cool to \(50^{\circ} \mathrm{C} ?\) Support your answer graphically.

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\ln \left(x^{2}+7\right)$$

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$4.1^{-\sqrt{3}}$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$0.8^{x}=4$$

For each statement, write an equivalent statement in exponential form. $$\log _{\sqrt{3}} 81=8$$

Newton's law of cooling says that the rate at which an object cools is proportional to the difference \(C\) in temperature between the object and the environment around it. The temperature \(f(t)\) of the object at time t in appropriate units after being introduced into an environment with a constant temperature \(T_{0}\) is $$f(t)=T_{0}+C e^{-k t}$$ where \(C\) and \(k\) are constants. Use this result. A piece of metal is heated to \(300^{\circ} \mathrm{C}\) and then placed in a cooling liquid at \(50^{\circ} \mathrm{C}\). After 4 minutes, the metal has cooled to \(175^{\circ} \mathrm{C}\). Estimate its temperature after 12 minutes.

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\ln \left(x^{4}+8\right)$$

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$6.4^{-\sqrt{3}}$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$0.6^{x}=3$$

For each statement, write an equivalent statement in exponential form. $$\log _{4} \frac{1}{64}=-3$$

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\ln \left(-x^{2}+4\right)$$

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$\sqrt{7} ^{\sqrt{7}}$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$4^{x-1}=3^{2 x}$$

For each statement, write an equivalent statement in exponential form. $$\log _{10} 0.001=-3$$

Emissions Governments could reduce carbon emissions by placing a tax on fossil fuels. The costbenefit equation $$\ln (1-P)=-0.0034-0.0053 x$$ estimates the relationship between a tax of \(x\) dollars per ton of carbon and the percent \(P\) reduction in emissions of carbon, where \(P\) is in decimal form. Determine \(P\) when \(x=60 .\) Interpret the result. (Source: Clime, W., The Economics of Global Warming, Institute for International Economics.)

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\ln \left(-x^{2}+16\right)$$

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$\sqrt{13}^{-\sqrt{13}}$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$2^{x+3}=5^{x}$$

For each statement, write an equivalent statement in exponential form. $$\log _{3} \sqrt[3]{9}=\frac{2}{3}$$

An amount \(A\) in a bank account after \(x\) years is given by \(A(x)=1000(1.025)^{x}\) (a) How much is in the account after 3 years? (b) How much is in the account after 10 years? (c) After how many years will there be about \(\$ 1900\) in the account?

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log _{4}\left(x^{2}-4 x-21\right)$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$6^{x+1}=4^{2 x-1}$$

Decide whether each function is one-to-one. Do not use a calculator. $$y=(x-2)^{2}$$

For each statement, write an equivalent statement in exponential form. $$\log \sqrt{10}=0.5$$

An amount \(A\) in a bank account after \(x\) years is given by \(A(x)=450(1.06)^{x}\). (a) How much is in the account after 2 years? (b) How much is in the account after 20 years? (c) After how many years will there be about \(\$ 2300\) in the account?

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log _{6}\left(2 x^{2}-7 x-4\right)$$

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$3^{x-4}=7^{2 x+5}$$

Decide whether each function is one-to-one. Do not use a calculator. $$y=-(x+3)^{2}-8$$

For each statement, write an equivalent statement in exponential form. $$\ln e^{6}=6$$

The concentration of bacteria \(B\) in millions per milliliter after \(x\) hours is given by $$B(x)=3.5 e^{0.02 x}$$ (a) How many bacteria are there after 1 hour? (b) How many bacteria are there after 6.5 hours? (c) After how many hours will there be 6 million bacteria per milliliter?

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log \left(x^{3}-x\right)$$

Decide whether each function is one-to-one. Do not use a calculator. $$y=2 x^{3}+1$$

Solve each equation. Give the exact answer. $$\log _{5} 125=x$$

The concentration of bacteria \(B\) in millions per milliliter after \(x\) hours is given by $$B(x)=1.33 e^{0.15 x}$$ (a) How many bacteria are there after 2.5 hours? (b) How many bacteria are there after 8 hours? (c) After how many hours will there be 31 million bacteria per milliliter?

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log \left(x^{3}-81 x\right)$$

Decide whether each function is one-to-one. Do not use a calculator. $$y=-2 x^{5}-4$$

Solve each equation. Give the exact answer. $$\log _{3} 81=x$$

Suppose that \(\$ 2500\) is invested in an account that pays interest compounded continuously. Find the amount of time that it would take for the account to grow to the given amount at the given rate of interest. Round to the nearest tenth of a year. \(\$ 3000\) at \(3.75 \%\)

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=-1.5^{x}$$

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log \left(\frac{x+3}{x-4}\right)$$