Chapter 12

Solve each equation. $$ \log _{4}\left(x^{2}-3 x\right)=1 $$

Write each as a logarithmic equation. $$ 5^{1 / 2}=\sqrt{5} $$

By inspection, find the value for \(x\) that makes each statement true. \(2^{x}=8\)

Find the exact value of each logarithm. $$ \log 0.0001 $$

If \(f(x)=3 x, g(x)=\sqrt{x}\), and \(h(x)=x^{2}+2,\) write each function as a composition with \(f, g\), or \(h .\) $$ F(x)=9 x^{2}+2 $$

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=\frac{1}{2} x-1 $$

Solve. $$ \frac{1}{4}=2^{3 x} $$

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

Solve each equation. $$ \log _{8}\left(x^{2}-2 x\right)=1 $$

Write each as a logarithmic equation. $$ 4^{1 / 3}=\sqrt[3]{4} $$

By inspection, find the value for \(x\) that makes each statement true. \(3^{x}=9\)

Find the exact value of each logarithm. $$ \log 0.001 $$

If \(f(x)=3 x, g(x)=\sqrt{x}\), and \(h(x)=x^{2}+2,\) write each function as a composition with \(f, g\), or \(h .\) $$ H(x)=3 x^{2}+6 $$

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=-\frac{1}{2} x+2 $$

Solve. $$ \frac{1}{27}=3^{2 x} $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 2 \log _{7} y+6 \log _{7} z $$

Solve each equation. $$ \log _{2} x+\log _{2}(3 x+1)=1 $$

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

By inspection, find the value for \(x\) that makes each statement true. \(5^{x}=\frac{1}{5}\)

Find the exact value of each logarithm. $$ \ln \sqrt{e} $$

If \(f(x)=3 x, g(x)=\sqrt{x}\), and \(h(x)=x^{2}+2,\) write each function as a composition with \(f, g\), or \(h .\) $$ G(x)=3 \sqrt{x} $$

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=x^{3} $$

Solve. $$ 9^{x}=27 $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{4} 2+\log _{4} 10-\log _{4} 5 $$

Solve each equation. $$ \log _{3} x+\log _{3}(x-8)=2 $$

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

By inspection, find the value for \(x\) that makes each statement true. \(4^{x}=1\)

Find the exact value of each logarithm. $$ \log \sqrt{10} $$

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=x^{3}-1 $$

Solve. $$ 32^{x}=4 $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{6} 21+\log _{6} 2-\log _{6} 7 $$

Solve. The size of the wolf population at Isle Royale National Park increases according to the formula \(y=y_{0} e^{0.043 t} .\) In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time 0 . If the size of the current population is 83 wolves, find how many there should be in 5 years. Round to the nearest whole number.

Find the value of each logarithmic expression. $$ \log _{2} \frac{1}{4} $$

An item is on sale for \(40 \%\) off its original price. If it is then marked down an additional \(60 \%,\) does this mean the item is free? Discuss why or why not.

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

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

Find the inverse of each one-to-one function. $$ f(x)=\frac{x-2}{5} $$

Solve. $$ 27^{x+1}=9 $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{7} 6+\log _{7} 3-\log _{7} 4 $$

Solve. The number of victims of a flu epidemic is increasing according to the formula \(y=y_{0} e^{0.075 t}\). In this formula, is time in weeks and \(y_{0}\) is the given population at time 0 . If 20,000 people are currently infected, how many might be infected in 3 weeks? Round to the nearest whole number.

Find the value of each logarithmic expression. $$ \log _{2} \frac{1}{32} $$

Uranium U-232 has a half-life of 72 years. What eventually happens to a 10 gram sample? Does it ever completely decay and disappear? Discuss why or why not.

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

Find the inverse of each one-to-one function. $$ f(x)=\frac{4 x-3}{2} $$

Solve. $$ 125^{x-2}=25 $$

Find \(f(x)\) and \(g(x)\) so that the given function \(h(x)=(f \circ g)(x)\). $$ h(x)=|x-1| $$

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{8} 5+\log _{8} 15-\log _{8} 20 $$

Solve. The population of the Cook Islands is decreasing according to the formula \(y=y_{0} e^{-0.0277 t}\). 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 \(11,870,\) use the formula to predict the population of Cook Islands in the year \(2025 .\) Round to the nearest whole number. (Source: The World Almanac)

Find the value of each logarithmic expression. $$ \log _{25} 5 $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ \ln x=1.4 $$