Chapter 9

Solve each equation or inequality. Check your solutions. $$ \log _{4} r=3 $$

The magnitude of an earthquake is measured on a logarithmic scale called the Richter scale. The magnitude \(M\) is given by \(M=\log _{10} x,\) where \(x\) represents the amplitude of the seismic wave causing ground motion. How many times as great was the motion caused by the 1906 San Francisco earthquake that measured 8.3 on the Richter scale as that caused by the 2001 Bhuj, India, earthquake that measured 6.9?

REVIEW If the equation \(y=3^{x}\) is graphed, which of the following values of \(x\) would produce a point closest to the \(x\) -axis? $$ \begin{array}{l}{\mathrm{F} \frac{3}{4}} \\ {\mathrm{G} \frac{1}{4}} \\\ {\mathrm{H} 0} \\ {\mathrm{J}-\frac{3}{4}}\end{array} $$

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. \(\log _{4} 68\)

A proposed city ordinance will make it illegal to create sound in a residential area that exceeds 72 decibels during the day and 55 decibels during the night. How many times as intense is the noise level allowed during the day than at night? (Hint: See information on page 514.)

Solve each equation. Check your solutions. $$ \frac{15}{p}+p=16 $$

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. \(\log _{6} 0.047\)

Solve each equation or inequality. Check your solutions. $$ \log _{3}(4 x-5)=5 $$

Consider the functions \(y=\log _{2} x+3, y=\log _{2} x-4, y=\log _{2}(x-1),\) and \(y=\log _{2}(x+2) .\) Use a graphing calculator to sketch the graphs on the same screen. Describe this family of graphs in terms of its parent graph \(y=\log _{2} x\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. \(\log _{50} 23\)

Use synthetic substitution to find \(f(-2)\) for \(f(x)=x^{3}+6 x-2\)

Solve each equation. Check your solutions. $$ \frac{2 a-5}{a-9}+\frac{a}{a+9}=\frac{-6}{a^{2}-81} $$

Solve each equation. Check your solutions. \(\log _{3}(a+3)+\log _{3}(a-3)=\log _{3} 16\)

Viviana has two dollars worth of nickels, dimes, and quarters. She has 18 total coins, and the number of nickels equals 25 minus twice the number of dimes. How many nickels, dimes, and quarters does she have?

Give an example of an exponential equation and its related logarithmic equation.

Identify each equation as a type of function. Then graph the equation. $$ y=\sqrt{x-2} $$

Solve each equation. Check your solutions. \(\log _{11} 2+2 \log _{11} x=\log _{11} 32\)

Write an equivalent exponential equation. $$ \log _{2} 3=x $$

Find the expression that does not belong. Explain. \(\log _{4} 16\) \(\qquad\) \(\log _{2} 16\) \(\qquad\) \(\log _{2} 4\) \(\qquad\) \(\log _{3} 9\)

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(m n=4\)

Write an equivalent exponential equation. $$ \log _{3} x=2 $$

Paul and Clemente are solving lo g 3 x = 9. Who is correct? Explain your reasoning. Paul \(\begin{aligned} \log _{3} x &=9 \\ 3^{x} &=9 \\ 3^{x} &=3^{2} \\ x &=2 \end{aligned}\) Clemente \(\begin{aligned} \log _{3} x &=9 \\ x &=3^{9} \\ x &=19,683 \end{aligned}\)

Identify each equation as a type of function. Then graph the equation. $$ y=8 $$

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(\frac{a}{b}=c\)

Write an equivalent exponential equation. $$ \log _{5} 125=3 $$

Using the definition of a logarithmic function where \(y=\log _{b} x\) explain why the base \(b\) cannot equal 1 .

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right] $$

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(y=-7 x\)

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{cc}{2} & {4} \\ {5} & {10}\end{array}\right] $$

Alexis has never scored a 3-point field goal, but she has scored a total of 59 points so far this season. She has made a total of 42 shots including free throws and 2-point field goals. How many free throws and 2-point field goals has Alexis scored?

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{cc}{-5} & {6} \\ {-11} & {3}\end{array}\right] $$

Solve each equation. Round to the nearest hundredth. \(2^{x}=10\)

What is the solution to the equation \(3^{x}=11 ?\) F. \(x=2\) G. \(x=\log _{10} 2\) H. \(x=\log _{10} 11+\log _{10} 3\) J. \(x=\frac{\log _{10} 11}{\log _{10} 3}\)

ENERGY A circular cell must deliver 18 watts of energy. If each square centimeter of the cell that is in sunlight produces 0.01 watt of energy, how long must the radius of the cell be?

Solve each equation. Round to the nearest hundredth. \(5^{x}=12\)

Simplify each expression. \(x^{\sqrt{6}} \cdot x^{\sqrt{6}}\)

Find \(g[h(x)]\) and \(h[g(x)]\) $$ \begin{array}{l}{h(x)=2 x-1} \\ {g(x)=x-5}\end{array} $$

Solve each equation. Round to the nearest hundredth. \(6^{x}=13\)

Simplify each expression. \(\left(b^{\sqrt{6}}\right)^{\sqrt{24}}\)

Find \(g[h(x)]\) and \(h[g(x)]\) $$ \begin{array}{l}{h(x)=x+3} \\ {g(x)=x^{2}}\end{array} $$

Solve each equation. Round to the nearest hundredth. \(2(1+0.1)^{x}=50\)

Solve each equation. Check your solutions. \(\frac{2 x+1}{x}-\frac{x+1}{x-4}=\frac{-20}{x^{2}-4 x}\)

Find \(g[h(x)]\) and \(h[g(x)]\) $$ \begin{array}{l}{h(x)=2 x+5} \\ {g(x)=-x+3}\end{array} $$

Solve each equation. Round to the nearest hundredth. \(10(1+0.25)^{x}=200\)

Solve each equation. Check your solutions. \(\frac{2 a-5}{a-9}-\frac{a-3}{3 a+2}=\frac{5}{3 a^{2}-25 a-18}\)

Solve each equation. Round to the nearest hundredth. \(400(1-0.2)^{x}=50\)

Solve each equation by using the method of your choice. Find exact solutions. \(9 y^{2}=49\)

Solve each equation by using the method of your choice. Find exact solutions. \(2 p^{2}=5 p+6\)

Donna Bowers has \(\$ 8000\) she wants to save in the bank. A 12 -month certificate of deposit (CD) earns 4\(\%\) annual interest, while a regular savings account earns 2\(\%\) annual interest. Ms. Bowers doesn't want to tie up all her money in a CD, but she has decided she wants to earn \(\$ 240\) in interest for the year. How much money should she put in to each type of account?

Simplify. Assume that no variable equals zero. \(x^{4} \cdot x^{6}\)