Chapter 12

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=\log x $$

Solve. Which of the following is the correct way to rewrite \(\log _{9} \frac{21}{3} ?\) a. \(\log _{9} 7\) b. \(\log _{9}(21-3)\) c. \(\frac{\log _{9} 21}{\log _{9} 3}\) d. \(\log _{9} 21-\log _{9} 3\)

Simplify. $$ \log _{9} 9 $$

Without using a calculator, explain which of \(\log 50\) or \(\ln 50\) must be larger.

Determine whether each statement is true or false. $$ \log _{2} x^{3}=3 \log _{2} x $$

Simplify. $$ \log _{2} 2 $$

Without using a calculator, explain which of \(\log 50^{-1}\) or \(\ln 50^{-1}\) must be larger.

Determine whether each statement is true or false. $$ \log _{3}(x+y)=\log _{3} x+\log _{3} y $$

Simplify. $$ \log _{8}(8)^{-1} $$

The Richter scale measures the intensity, or magnitude, of an earthquake. The formula for the magnitude \(R\) of an earthquake is \(R=\log \left(\frac{a}{T}\right)+B\), where a is the amplitude in micrometers of the vertical motion of the ground at the recording station, \(T\) is the number of seconds between successive seismic waves, and \(B\) is an adjustment factor that takes into account the weakening of the seismic wave as the distance increases from the epicenter of the earthquake. Use the Richter scale formula to find the magnitude \(R\) of the earthquake that fits the description given. Round answers to one decimal place. Amplitude \(a\) is 200 micrometers, time \(T\) between waves is 1.6 seconds, and \(B\) is 2.1 .

Determine whether each statement is true or false. $$ \frac{\log _{7} 10}{\log _{7} 5}=\log _{7} 2 $$

Simplify. $$ \log _{11}(11)^{-1} $$

The Richter scale measures the intensity, or magnitude, of an earthquake. The formula for the magnitude \(R\) of an earthquake is \(R=\log \left(\frac{a}{T}\right)+B\), where a is the amplitude in micrometers of the vertical motion of the ground at the recording station, \(T\) is the number of seconds between successive seismic waves, and \(B\) is an adjustment factor that takes into account the weakening of the seismic wave as the distance increases from the epicenter of the earthquake. Use the Richter scale formula to find the magnitude \(R\) of the earthquake that fits the description given. Round answers to one decimal place. Amplitude \(a\) is 150 micrometers, time \(T\) between waves is 3.6 seconds, and \(B\) is 1.9 .

Determine whether each statement is true or false. $$ \log _{7} \frac{14}{8}=\log _{7} 14-\log _{7} 8 $$

Graph each logarithmic function. $$ y=\log _{3} x $$

The Richter scale measures the intensity, or magnitude, of an earthquake. The formula for the magnitude \(R\) of an earthquake is \(R=\log \left(\frac{a}{T}\right)+B\), where a is the amplitude in micrometers of the vertical motion of the ground at the recording station, \(T\) is the number of seconds between successive seismic waves, and \(B\) is an adjustment factor that takes into account the weakening of the seismic wave as the distance increases from the epicenter of the earthquake. Use the Richter scale formula to find the magnitude \(R\) of the earthquake that fits the description given. Round answers to one decimal place. Amplitude \(a\) is 400 micrometers, time \(T\) between waves is 2.6 seconds, and \(B\) is 3.1

Determine whether each statement is true or false. $$ \frac{\log _{7} x}{\log _{7} y}=\log _{7} x-\log _{7} y $$

Graph each logarithmic function. $$ y=\log _{2} x $$

The Richter scale measures the intensity, or magnitude, of an earthquake. The formula for the magnitude \(R\) of an earthquake is \(R=\log \left(\frac{a}{T}\right)+B\), where a is the amplitude in micrometers of the vertical motion of the ground at the recording station, \(T\) is the number of seconds between successive seismic waves, and \(B\) is an adjustment factor that takes into account the weakening of the seismic wave as the distance increases from the epicenter of the earthquake. Use the Richter scale formula to find the magnitude \(R\) of the earthquake that fits the description given. Round answers to one decimal place. Amplitude \(a\) is 450 micrometers, time \(T\) between waves is 4.2 seconds, and \(B\) is 2.7 .

Determine whether each statement is true or false. $$ \left(\log _{3} 6\right) \cdot\left(\log _{3} 4\right)=\log _{3} 24 $$

Graph each logarithmic function. $$ f(x)=\log _{1 / 4} x $$

Graph each logarithmic function. $$ f(x)=\log _{1 / 2} x $$

Graph each logarithmic function. $$ f(x)=\log _{5} x $$

Graph each logarithmic function. $$ f(x)=\log _{6} x $$

Graph each logarithmic function. $$ f(x)=\log _{1 / 6} x $$

Graph each logarithmic function. $$ f(x)=\log _{1 / 5} x $$

Simplify each rational expression. $$ \frac{x+3}{3+x} $$

Simplify each rational expression. $$ \frac{x-5}{5-x} $$

Simplify each rational expression. $$ \frac{x^{2}-8 x+16}{2 x-8} $$

Let \(f(x)=\log _{5} x\). Then \(g(x)=5^{x}\) is the inverse of \(f(x)\). The ordered pair (2,25) is a solution of the function \(g(x)\). a. Write this solution using function notation. b. Write an ordered pair that we know to be a solution of \(f(x)\) c. Use the answer to part (b) and write the solution using function notation.

Let \(f(x)=\log _{0.3} x\). Then \(g(x)=0.3^{x}\) is the inverse of \(f(x)\). The ordered pair (3,0.027) is a solution of the function \(g(x)\). a. Write this solution using function notation. b. Write an ordered pair that we know to be a solution of \(f(x)\). c. Use the answer to part (b) and write the solution using function notation.

Explain why negative numbers are not included as logarithmic bases.

Explain why 1 is not included as a logarithmic base.

Graph each function and its inverse on the same set of axes. $$ y=4^{x} ; y=\log _{4} x $$

Graph each function and its inverse on the same set of axes. $$ y=3^{x} ; y=\log _{3} x $$

Graph each function and its inverse on the same set of axes. $$ y=\left(\frac{1}{3}\right)^{x} ; y=\log _{1 / 3} x $$

Graph each function and its inverse on the same set of axes. $$ y=\left(\frac{1}{2}\right)^{x} ; y=\log _{1 / 2} x $$

Explain why the graph of the function \(y=\log _{b} x\) contains the point (1,0) no matter what \(b\) is.

\(\log _{3} 10\) is between which two integers? Explain your answer.

The formula \(\log _{10}(1-k)=\frac{-.3}{H}\) models the relationship between the half-life \(H\) of a radioactive material and its rate of decay \(k\). Find the rate of decay of the iodine isotope \(I-131\) if its half-life is 8 days. Round to four decimal places.

The formula \(\mathrm{pH}=-\log _{10}\left(\mathrm{H}^{+}\right)\) gives the \(\mathrm{pH}\) for a liquid, where \(\mathrm{H}^{+}\) stands for the concentration of hydronium ions. Find the \(\mathrm{pH}\) of lemonade, whose concentration of hydronium ions is 0.0050 moles/liter.