Chapter 12

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{7} \frac{5 x}{4} $$

Find the value of each logarithmic expression. $$ \log _{3} 81 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln 4 x=0.18 $$

Solve each equation for \(y .\) $$ x=-2 y-7 $$

Solve. Unless otherwise indicated, round results to one decimal place. The equation \(y=84,949(1.096)^{x}\) models the number of American college students who studied abroad each year from 1995 through \(2006 .\) In the equation, \(y\) is the number of American students studying abroad and \(x\) represents the number of years after \(1995 .\) Round answers to the nearest whole. (Source: Based on data from Institute of International Education, Open Doors 2006 ) a. Estimate the number of American students studying abroad in 2000 . b. Assuming this equation continues to be valid in the future, use this equation to predict the number of American students studying abroad in 2020 .

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{4} \frac{2}{9 z} $$

The formula \(w=0.00185 h^{2.67}\) is used to estimate the normal weight w in pounds of a boy h inches tall. Use this formula to solve. Round to the nearest tenth. Find the expected height of a boy who weighs 140 pounds.

Find the value of each logarithmic expression. $$ \log _{2} 16 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln 3 x=0.76 $$

Solve each equation for \(y .\) $$ x=4 y+7 $$

Solve. Unless otherwise indicated, round results to one decimal place. Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) is a greenhouse gas that contributes to global warming. Partially due to the combustion of fossil fuels, the amount of \(\mathrm{CO}_{2}\) in Earth's atmosphere has been increasing by \(0.4 \%\) annually over the past century. In \(2000,\) the concentration of \(\mathrm{CO}_{2}\) in the atmosphere was 369.4 parts per million by volume. To make the following predictions, use \(y=369.4(1.004)^{t}\) where \(y\) is the concentration of \(\mathrm{CO}_{2}\) in parts per million and \(t\) is the number of years after 2000. (Sources: Based on data from the United Nations Environment Programme and the Carbon Dioxide Information Analysis Center) a. Predict the concentration of \(\mathrm{CO}_{2}\) in the atmosphere in the year 2012 . b. Predict the concentration of \(\mathrm{CO}_{2}\) in the atmosphere in the year 2030 .

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{9} \frac{7}{8 y} $$

The formula \(P=14.7 e^{-0.21 x}\) gives the average atmospheric pressure \(P\), in pounds per square inch, at an altitude \(x\), in miles above sea level. Use this formula to solve Exercises 43 through 46. Round to the nearest tenth. Find the average atmospheric pressure of Denver, which is 1 mile above sea level.

Find the value of each logarithmic expression. $$ \log _{4} \frac{1}{64} $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln (3 x-4)=2.3 $$

Business people are concerned with cost functions, revenue functions, and profit functions. Recall that the profit \(P(x)\) obtained from selling \(x\) units of a product is equal to the revenue \(R(x)\) from selling the \(x\) units minus the cost \(C(x)\) of manufacturing the \(x\) units. Write an equation expressing this relationship among \(C(x), R(x),\) and \(P(x)\)

Evaluate each exponential expression. $$ 25^{1 / 2} $$

The equation \(y=136.76(1.115)^{x}\) gives the number of cellular phone users y (in millions) in the United States for the years 2002 through \(2009 .\) In this equation \(x=0\) corresponds to \(2002, x=1\) corresponds to \(2003,\) and so on. Use this model to solve Exercises 43 and \(44 .\) Round answers to the nearest tenth of a million.Predict the number of cell phone users in the year 2012 .

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{2} \frac{x^{3}}{y} $$

The formula \(P=14.7 e^{-0.21 x}\) gives the average atmospheric pressure \(P\), in pounds per square inch, at an altitude \(x\), in miles above sea level. Use this formula to solve. Round to the nearest tenth. Find the average atmospheric pressure of Pikes Peak, which is 2.7 miles above sea level.

Find the value of each logarithmic expression. $$ \log _{3} \frac{1}{9} $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln (2 x+5)=3.4 $$

Suppose the revenue \(R(x)\) for \(x\) units of a product can be described by \(R(x)=25 x\), and the cost \(C(x)\) can be described by \(C(x)=50+x^{2}+4 x\). Find the profit \(P(x)\) for \(x\) units.

Evaluate each exponential expression. $$ 49^{1 / 2} $$

The equation \(y=136.76(1.115)^{x}\) gives the number of cellular phone users y (in millions) in the United States for the years 2002 through \(2009 .\) In this equation \(x=0\) corresponds to \(2002, x=1\) corresponds to \(2003,\) and so on. Use this model to solve Exercises 43 and \(44 .\) Round answers to the nearest tenth of a million. Predict the number of cell phone users in 2014 .

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{5} \frac{x}{y^{4}} $$

The formula \(P=14.7 e^{-0.21 x}\) gives the average atmospheric pressure \(P\), in pounds per square inch, at an altitude \(x\), in miles above sea level. Use this formula to solve. Round to the nearest tenth. Find the elevation of a Delta jet if the atmospheric pressure outside the jet is 7.5 pounds per square inch.

Solve. \(\log _{3} 9=x\)

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \log (2 x+1)=-0.5 $$

If you are given \(f(x)\) and \(g(x)\), explain in your own words how to find \((f \circ g)(x),\) and then how to find \((g \circ f)(x)\)

Evaluate each exponential expression. $$ 16^{3 / 4} $$

Solve. Use \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) Round answers to two decimal places. Find the amount a college student owes at the end of 3 years if \(\$ 6000\) is loaned to her at a rate of \(8 \%\) compounded monthly.

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{b} \sqrt{7 x} $$

The formula \(P=14.7 e^{-0.21 x}\) gives the average atmospheric pressure \(P\), in pounds per square inch, at an altitude \(x\), in miles above sea level. Use this formula to solve. Round to the nearest tenth. Find the elevation of a remote Himalayan peak if the atmospheric pressure atop the peak is 6.5 pounds per square inch.

Solve. $$ \log _{2} 8=x $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \log (3 x-2)=-0.8 $$

Given \(f(x)\) and \(g(x),\) describe in your own words the difference between \((f \circ g)(x)\) and \((f \cdot g)(x)\).

Evaluate each exponential expression. $$ 49^{1 / 2} $$

Solve. Use \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) Round answers to two decimal places. Find the amount owed at the end of 5 years if \(\$ 3000\) is loaned at a rate of \(10 \%\) compounded quarterly.

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{b} \sqrt{\frac{3}{y}} $$

Psychologists call the graph of the formula \(t=\frac{1}{c} \ln \left(\frac{A}{A-N}\right)\) the learning curve since the formula relates time t passed, in weeks, to a measure \(N\) of learning achieved, to a measure \(A\) of maximum learning possible, and to a measure c of an individual's learning style. Use this formula to answer Exercises 47 through \(50 .\) Round to the nearest whole number. Norman Weidner is learning to type. If he wants to type at a rate of 50 words per minute \((N=50)\) and his expected maximum rate is 75 words per minute \((A=75)\), how many weeks should it take him to achieve his goal? Assume that \(c\) is 0.09 .

Solve. $$ \log _{3} x=4 $$

Use the formula \(A=P e^{r t}\) to solve. How much money does Dana Jones have after 12 years if she invests \(\$ 1400\) at \(8 \%\) interest compounded continuously?

Evaluate each exponential expression. $$ 9^{-3 / 2} $$

Solve. Use \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) Round answers to two decimal places. Find the total amount a college student has in a savings account if \(\$ 2000\) was invested and earned \(6 \%\) compounded semiannually for 12 years.

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{6} x^{4} y^{5} $$

Psychologists call the graph of the formula \(t=\frac{1}{c} \ln \left(\frac{A}{A-N}\right)\) the learning curve since the formula relates time t passed, in weeks, to a measure \(N\) of learning achieved, to a measure \(A\) of maximum learning possible, and to a measure c of an individual's learning style. Use this formula to answer. Round to the nearest whole number. An experiment of teaching chimpanzees sign language shows that a typical chimp can master a maximum of 65 signs. How many weeks should it take a chimpanzee to master 30 signs if \(c\) is 0.03 ?

Solve. $$ \log _{2} x=3 $$

Use the formula \(A=P e^{r t}\) to solve. Determine the size of an account in which \(\$ 3500\) earns \(6 \%\) interest compounded continuously for 1 year.

Evaluate each exponential expression. $$ 81^{-3 / 4} $$