Chapter 12

Find \(f(x)\) and \(g(x)\) so that the given function \(h(x)=(f \circ g)(x)\). $$ h(x)=\sqrt{x+5}+2 $$

Find the inverse of each one-to-one function. $$ f(x)=\sqrt[3]{x} $$

Solve. $$ 81^{x-1}=27^{2 x} $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{10} x-\log _{10}(x+1)+\log _{10}\left(x^{2}-2\right) $$

Solve. The population of Saint Barthelemy is decreasing according to the formula \(y=y_{0} e^{-0.0034 t} .\) In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time 0 . If the size of the population in 2009 was 7448 , use the formula to predict the population of Saint Barthelemy in the year 2025 . Round to the nearest whole number. (Source: The World Almanac)

Find the value of each logarithmic expression. $$ \log _{8} \frac{1}{2} $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln x=2.1 $$

Find \(f(x)\) and \(g(x)\) so that the given function \(h(x)=(f \circ g)(x)\). $$ h(x)=(3 x+4)^{2}+3 $$

Find the inverse of each one-to-one function. $$ f(x)=\sqrt[3]{x+1} $$

Solve. $$ 4^{3 x-7}=32^{2 x} $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{9}(4 x)-\log _{9}(x-3)+\log _{9}\left(x^{3}+1\right) $$

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve these compound interest problems. Round to the nearest tenth. How long does it take for \(\$ 600\) to double if it is invested at \(7 \%\) interest compounded monthly?

Find the value of each logarithmic expression. $$ \log _{1 / 2} 2 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \log x=2.3 $$

Find the inverse of each one-to-one function. $$ f(x)=\frac{5}{3 x+1} $$

Solve. $$ \left(\frac{1}{8}\right)^{x}=16^{1-x} $$

Find \(f(x)\) and \(g(x)\) so that the given function \(h(x)=(f \circ g)(x)\). $$ h(x)=\frac{1}{2 x-3} $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 3 \log _{2} x+\frac{1}{2} \log _{2} x-2 \log _{2}(x+1) $$

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve these compound interest problems. Round to the nearest tenth. How long does it take for \(\$ 600\) to double if it is invested at \(12 \%\) interest compounded monthly?

Find the value of each logarithmic expression. $$ \log _{2 / 3} \frac{4}{9} $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \log x=3.1 $$

Find \(f(x)\) and \(g(x)\) so that the given function \(h(x)=(f \circ g)(x)\). $$ h(x)=\frac{1}{x+10} $$

Find the inverse of each one-to-one function. $$ f(x)=\frac{7}{2 x+4} $$

Solve. $$ \left(\frac{1}{9}\right)^{x}=27^{2-x} $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 2 \log _{5} x+\frac{1}{3} \log _{5} x-3 \log _{5}(x+5) $$

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve these compound interest problems. Round to the nearest tenth. How long does it take for a \(\$ 1200\) investment to earn \(\$ 200\) interest if it is invested at \(9 \%\) interest compounded quarterly?

Find the value of each logarithmic expression. $$ \log _{6} 1 $$

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

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

Find the inverse of each one-to-one function. $$ f(x)=(x+2)^{3} $$

Objective C) Solve. Unless otherwise indicated, round results to one decimal place. See Example 7.One type of uranium has a radioactive decay rate of \(0.4 \%\) per day. If 30 pounds of this uranium is available today, how much will still remain after 50 days? Use \(y=30(0.996)^{x},\) and let \(x\) be 50

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 2 \log _{8} x-\frac{2}{3} \log _{8} x+4 \log _{8} x $$

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

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln x=-3.7 $$

Solve each equation for \(y .\) $$ x=y-5 $$

Find the inverse of each one-to-one function. $$ f(x)=(x-5)^{3} $$

Solve. Unless otherwise indicated, round results to one decimal place. The nuclear waste from an atomic energy plant decays at a rate of \(3 \%\) each century. If 150 pounds of nuclear waste is disposed of, how much of it will still remain after 10 centuries? Use \(y=150(0.97)^{x}\), and let \(x\) be 10

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve these compound interest problems. Round to the nearest tenth. How long does it take for a \(\$ 1500\) investment to earn \(\$ 200\) interest if it is invested at \(10 \%\) interest compounded semiannually?

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 5 \log _{6} x-\frac{3}{4} \log _{6} x+3 \log _{6} x $$

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve these compound interest problems. Round to the nearest tenth. How long does it take for \(\$ 1000\) to double if it is invested at \(8 \%\) interest compounded semiannually?

Find the value of each logarithmic expression. $$ \log _{10} 100 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \log 2 x=1.1 $$

Solve each equation for \(y .\) $$ x=3 y $$

Solve. Unless otherwise indicated, round results to one decimal place. Cheese production in the United States is currently growing at a rate of \(3 \%\) per year. The equation \(y=8.6(1.03)^{x}\) models the cheese production in the United States from 2003 to \(2009 .\) In this equation, \(y\) is the amount of cheese produced, in billions of pounds, and \(x\) represents the number of years after 2003 . Round answers to the nearest tenth of a billion. (Source: National Agricultural Statistics Service) a. Estimate the total cheese production in the United States in 2007 . b. Assuming this equation continues to be valid in the future, use the equation to predict the total amount of cheese produced in the United States in \(2015 .\)

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{3} \frac{4 y}{5} $$

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve these compound interest problems. Round to the nearest tenth. How long does it take for \(\$ 1000\) to double if it is invested at \(8 \%\) interest compounded monthly?

Find the value of each logarithmic expression. $$ \log _{10} \frac{1}{10} $$

Solve each equation for \(y .\) $$ x=-6 y $$

Solve. Unless otherwise indicated, round results to one decimal place. Retail revenue from shopping on the Internet is currently growing at rate of \(26 \%\) per year. In \(2003,\) a total of \(\$ 39\) billion in revenue was collected through Internet retail sales. Answer the following questions using \(y=39(1.26)^{t},\) where \(y\) is Internet revenues in billions of dollars and \(t\) is the number of years after 2003\. Round answers to the nearest tenth of a billion dollars. (Source: U.S. Bureau of the Census) a. According to the model, what level of retail revenues from Internet shopping was expected in \(2005 ?\) b. If the given model continues to be valid, predict the level of Internet shopping revenues in 2012 .

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \log 3 x=1.3 $$