Chapter 5

Graph each equation. $$f(x)=3^{x}$$

Solve each equation. Express all solutions in exact form. $$3 e^{2 x}+1=5$$

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=-3 x+5$$

In Exercises 1 and \(2,\) match the logarithm in Column I with its value in Column II. Remember that \(\log _{a} x\) is the exponent to which a must be raised in order to obtain \(x\). (a) \(\log _{2} 16\) (b) \(\log _{3} 1\) (c) \(\log _{10} 0.1\) (d) \(\log _{2} \sqrt{2}\) (e) \(\log _{e} \frac{1}{e^{2}}\) (f) \(\log _{1 / 2} 8\) A. 0 B. \(\frac{1}{2}\) C. 4 D. \(-3\) E. \(-1\) F. \(-2\)

The information allows us to use the function \(A(t)=A_{0} e^{-0.0001216}\) to approximate the amount of carbon 14 remaining in a sample, where \(t\) is in years. Use this function (Note: \(-0.0001216 \approx-\frac{\ln 2}{5700}\) ) Dating Suppose an Egyptian mummy is discovered in which the amount of carbon 14 present is only about one-third the amount found in the atmosphere. About how long ago did the Egyptian die?

The information allows us to use the function \(A(t)=A_{0} e^{-0.0001216}\) to approximate the amount of carbon 14 remaining in a sample, where \(t\) is in years. Use this function (Note: \(-0.0001216 \approx-\frac{\ln 2}{5700}\) ) A sample from a refuse deposit near the Strait of Magellan had \(60 \%\) of the carbon 14 of a contemporary living sample. Estimate the age of the sample.

Graph each equation. $$f(x)=4^{x}$$

Solve each equation. Express all solutions in exact form. $$\frac{1}{2} e^{x}=13$$

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=-5 x+2$$

In Exercises 1 and \(2,\) match the logarithm in Column I with its value in Column II. Remember that \(\log _{a} x\) is the exponent to which a must be raised in order to obtain \(x\). (a) \(\log _{3} 81\) (b) \(\log _{3} \frac{1}{3}\) (c) \(\log _{10} 0.01\) (d) \(\log _{6} \sqrt{6}\) (e) \(\log _{e} 1\) (f) \(\log _{3} 27^{3 / 2}\) A. \(-2\) B. \(-1\) C. 0 D. \(\frac{1}{2}\) E. \(\frac{9}{2}\) F. 4

Graph each equation. $$f(x)=\left(\frac{1}{3}\right)^{x}$$

Solve each equation. Express all solutions in exact form. $$2\left(10^{x}\right)=14$$

For each statement, write an equivalent statement in logarithmic form. $$3^{4}=81$$

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

Graph each equation. $$f(x)=\left(\frac{1}{4}\right)^{x}$$

Solve each equation. Express all solutions in exact form. $$5\left(10^{3 x}\right)-4=6$$

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

For each statement, write an equivalent statement in logarithmic form. $$2^{5}=32$$

The information allows us to use the function \(A(t)=A_{0} e^{-0.0001216}\) to approximate the amount of carbon 14 remaining in a sample, where \(t\) is in years. Use this function (Note: \(-0.0001216 \approx-\frac{\ln 2}{5700}\) ) Estimate the age of a specimen that contains \(20 \%\) of the carbon 14 of a comparable living specimen.

The half-life of radioactive lead 210 is 21.7 years. (a) Find an exponential decay model for lead 210 . (b) Estimate how long it will take a sample of 500 grams to decay to 400 grams. (c) Estimate how much of the sample of 500 grams will remain after 10 years.

Solve each equation. $$4^{x}=2$$

Solve each equation. Express all solutions in exact form. $$\frac{1}{2} \log _{2} x=\frac{3}{4}$$

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=\sqrt{36-x^{2}}$$

For each statement, write an equivalent statement in logarithmic form. $$\left(\frac{1}{2}\right)^{-4}=16$$

Radioactive cesium 137 was emitted in large amounts in the Chernobyl nuclear power station accident in Russia on April \(26,1986 .\) The amount of cesium 137 remaining after \(x\) years in an initial sample of 100 milligrams can be described by $$A(x)=100 e^{-0.02295 x}$$ (Source: Mason, C., Biology of Freshwater Pollution, John Wiley and Sons.) (a) Estimate how much is remaining after 50 years. Is the half-life of cesium 137 greater or less than 50 years? (b) Estimate the half-life of cesium 137 .

Solve each equation. $$125^{x}=5$$

Solve each equation. Express all solutions in exact form. $$2 \log _{3} x=\frac{4}{5}$$

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=-\sqrt{100-x^{2}}$$

For each statement, write an equivalent statement in logarithmic form. $$\left(\frac{2}{3}\right)^{-3}=\frac{27}{8}$$

The table shows the amount \(y\) of polonium 210 remaining after \(t\) days from an initial sample of 2 milligrams. $$\begin{array}{ll|l|l|l}t \text { (days) } & 0 & 100 & 200 & 300 \\\\\hline y \text { (milligrams) } & 2 & 1.22 & 0.743 & 0.453\end{array}$$ (a) Use the table to determine whether the half-life of polonium 210 is greater or less than 200 days. (b) Find a formula that models the amount \(A\) of polonium 210 in the table after \(t\) days. (c) Estimate the half-life of polonium 210 .

Each function f is an exponential function. Therefore, each function f^{-1} is a(n)______ function.

Solve each equation. $$\left(\frac{1}{2}\right)^{x}=4$$

Solve each equation. Express all solutions in exact form. $$4 \ln 3 x=8$$

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

For each statement, write an equivalent statement in logarithmic form. $$10^{-4}=0.0001$$

Sound Intensity Use the formula $$d=10 \log \frac{I}{I_{0}}$$ to estimate the average decibel level for each sound with the given intensity \(I .\) For comparison, conversational speech has a sound level of about 60 decibels. (a) Jackhammer: \(31,620,000,000 I_{0}\). (b) iPhone 5 speakers: \(10^{1 /} l_{0}\). (c) Rock singer screaming into microphone: \(10^{14} I_{0}\).

Solve each equation. $$\left(\frac{2}{3}\right)^{x}=\frac{9}{4}$$

Solve each equation. Express all solutions in exact form. $$7 \ln 2 x=10$$

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=\sqrt[3]{x}$$

For each statement, write an equivalent statement in logarithmic form. $$\left(\frac{1}{100}\right)^{-2}=10,000$$

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$2^{\sqrt{10}}$$

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. $$3^{x}=7$$

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=|2 x+1|$$

For each statement, write an equivalent statement in logarithmic form. $$e^{0}=1$$

The earthquake off the coast of Northern Sumatra on Dec. \(26,2004,\) had a Richter scale rating of 8.9 (a) Express the intensity of this earthquake in terms of \(I_{0}\). (b) Aftershocks from this quake had a Richter scale rating of \(6.0 .\) Express the intensity of these in terms of \(I_{0}\) (c) Compare the intensities of the 8.9 earthquake to the 6.0 aftershock.

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

Use a calculator to find an approximation for each power. Give the maximum number of decimal places that your calculator displays. $$3^{\sqrt{11}}$$

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. $$5^{x}=13$$

Can a quadratic function \(f\) with domain \((-\infty, \infty)\) have an inverse function? Explain.

For each statement, write an equivalent statement in logarithmic form. $$e^{1 / 3}=\sqrt[3]{e}$$