Chapter 12

Given the one-to-one function \(f(x)=x^{3}+2,\) find the following. (Hint: You do not need to find the equation for \(f^{-1}\).) a. \(f(-1)\) b. \(f^{-1}(1)\)

If \(f(x)=x^{2}-6 x+2, g(x)=-2 x\), and \(h(x)=\sqrt{x}\), find each composition. $$ (g \circ h)(0) $$

Graph each exponential function. $$ y=-\left(\frac{1}{4}\right)^{x} $$

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{2} x-\log _{2} y $$

Solve each equation. Give an exact solution and a four-decimal-place approximation. $$ 5^{2 x-6}=12 $$

Write each as an exponential equation. $$ \log _{1.2} 1.44=2 $$

Solve. Unless noted otherwise, round answers to the nearest whole. Suppose a city with population 320,000 has been growing at a rate of \(4 \%\) per year. If this rate continues, find the population of this city in 20 years.

Use a calculator to approximate each logarithm to four decimal places. $$ \ln 41.5 $$

Given the one-to-one function \(f(x)=x^{3}+2,\) find the following. (Hint: You do not need to find the equation for \(f^{-1}\).) a. \(f(-2)\) b. \(f^{-1}(-6)\)

If \(f(x)=x^{2}-6 x+2, g(x)=-2 x\), and \(h(x)=\sqrt{x}\), find each composition. $$ (h \circ g)(0) $$

Graph each exponential function. $$ y=-\left(\frac{1}{5}\right)^{x} $$

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{3} 12-\log _{3} z $$

Solve each equation. $$ \log _{2}(x+5)=4 $$

Write each as an exponential equation. $$ \log _{3} \frac{1}{81}=-4 $$

Solve. Unless noted otherwise, round answers to the nearest whole. The number of employees for a certain company has been decreasing each year by \(5 \%\). If the company currently has 640 employees and this rate continues, find the number of employees in 10 years.

Find the exact value of each logarithm. $$ \log 100 $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=x^{2}+1 ; g(x)=5 x $$

Graph each exponential function. $$ y=\left(\frac{1}{3}\right)^{x}+1 $$

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{2}\left(x^{2}+6\right)-\log _{2}\left(x^{2}+1\right) $$

Solve each equation. $$ \log _{2}(x-5)=3 $$

Write each as an exponential equation. $$ \log _{1 / 4} 16=-2 $$

Solve. Unless noted otherwise, round answers to the nearest whole. The number of students attending summer school at a local community college has been decreasing each year by \(7 \% .\) If 984 students currently attend summer school and this rate continues, find the number of students attending summer school in 5 years.

Find the exact value of each logarithm. $$ \log 10,000 $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=x-3 ; g(x)=x^{2} $$

Graph each exponential function. $$ y=\left(\frac{1}{2}\right)^{x}+2 $$

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{7}(x+9)-\log _{7}\left(x^{2}+10\right) $$

Solve each equation. $$ \log _{4} 2+\log _{4} x=0 $$

Write each as a logarithmic equation. $$ 2^{4}=16 $$

Solve. Unless noted otherwise, round answers to the nearest whole. National Park Service personnel are trying to increase the size of the bison population of Theodore Roosevelt National Park. If 260 bison currently live in the park, and if the population's rate of growth is \(2.5 \%\) annually, find how many bison there should be in 10 years.

Find the exact value of each logarithm. $$ \log \frac{1}{1000} $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=2 x-3 ; g(x)=x+7 $$

Graph each exponential function. $$ f(x)=2^{x-2} $$

Use the power property to rewrite each expression. $$ \log _{3} x^{2} $$

Solve each equation. $$ \log _{3} 5+\log _{3} x=1 $$

Write each as a logarithmic equation. $$ 5^{3}=125 $$

Solve. Unless noted otherwise, round answers to the nearest whole. The size of the rat population of a wharf area grows at a rate of \(8 \%\) monthly. If there are 200 rats in January, find how many rats should be expected by next January.

Find the exact value of each logarithm. $$ \log \frac{1}{10} $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=x+10 ; g(x)=3 x+1 $$

Graph each exponential function. $$ g(x)=2^{x+1} $$

Use the power property to rewrite each expression. $$ \log _{2} x^{5} $$

Solve each equation. $$ \log _{2} 6-\log _{2} x=3 $$

Write each as a logarithmic equation. $$ 10^{2}=100 $$

Solve. Unless noted otherwise, round answers to the nearest whole. A rare isotope of a nuclear material is very unstable, decaying at a rate of \(15 \%\) each second. Find how much isotope remains 10 seconds after 5 grams of the isotope is created.

Find the exact value of each logarithm. $$ \ln e^{2} $$

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=x^{3}+x-2 ; g(x)=-2 x $$

Graph each exponential function. $$ F(x)=5^{x+1} $$

Use the power property to rewrite each expression. $$ \log _{4} 5^{-1} $$

Solve each equation. $$ \log _{4} 10-\log _{4} x=2 $$

Write each as a logarithmic equation. $$ 10^{4}=10,000 $$

Solve. Unless noted otherwise, round answers to the nearest whole. An accidental spill of 75 grams of radioactive material in a local stream has led to the presence of radioactive debris decaying at a rate of \(4 \%\) each day. Find how much debris still remains after 14 days.