Chapter 9

EARTHQUAKES The great Alaskan earthquake, in \(1964,\) was about 100 times as intense as the Loma Prieta earthquake in San Francisco, in \(1989 .\) Find the difference in the Richter scale magnitudes of the earthquakes.

The 1971 Sylmar earthquake in Los Angeles had a Richter scale magnitude of \(6.3 .\) Suppose an architect has designed a building strong enough to withstand an earthquake 50 times as intense as the Sylmar quake. Find the magnitude of the strongest quake this building can withstand.

Sketch the graph of each function. Then state the function's domain and range. $$ y=4\left(\frac{1}{3}\right)^{x} $$

Solve each inequality. $$ \log _{2}(3 x-5)>\log _{2}(x+7) $$

OPEN ENDED Give an example of a quantity that grows or decays at a fixed rate. Write a real-world problem involving the rate and solve by using logarithms.

Use a calculator to evaluate each expression to four decimal places. \(\ln 10\)

PROBABILITY For Exercises \(22-24,\) use the following information. In the 1930 \(\mathrm{s}\) . Dr. Frank Benford demonstrated a way to determine whether a set of numbers have been randomly chosen or the numbers have been manually chosen. If the sets of numbers were not randomly chosen, then the Benford formula, \(P=\log _{10}\left(1+\frac{1}{d}\right),\) predicts the probability of a digit \(d\) being the first digit of the set. For example, there is a 4.6\(\%\) probability that the first digit is \(9 .\) Rewrite the formula to solve for the digit if given the probability.

Solve each equation or inequality. Round to four decimal places. $$ 5^{x}=52 $$

Determine whether each function represents exponential growth or decay. $$ y=10(3.5)^{x} $$

Solve each inequality. Check your solutions. \(\log _{5}(5 x-7) \leq \log _{5}(2 x+5)\)

Use a calculator to evaluate each expression to four decimal places. \(\ln 5.42\)

PROBABILITY For Exercises \(22-24,\) use the following information. In the 1930 \(\mathrm{s}\) . Dr. Frank Benford demonstrated a way to determine whether a set of numbers have been randomly chosen or the numbers have been manually chosen. If the sets of numbers were not randomly chosen, then the Benford formula, \(P=\log _{10}\left(1+\frac{1}{d}\right),\) predicts the probability of a digit \(d\) being the first digit of the set. For example, there is a 4.6\(\%\) probability that the first digit is \(9 .\) Find the digit that has a 9.7\(\%\) probability of being selected.

Solve each equation or inequality. Round to four decimal places. $$ 4^{3 p}=10 $$

Determine whether each function represents exponential growth or decay. $$ y=2(4)^{x} $$

Write each equation in exponential form. \(\log _{5} 125=3\)

Use a calculator to evaluate each expression to four decimal places. \(\ln 0.03\)

PROBABILITY For Exercises \(22-24,\) use the following information. In the 1930 \(\mathrm{s}\) . Dr. Frank Benford demonstrated a way to determine whether a set of numbers have been randomly chosen or the numbers have been manually chosen. If the sets of numbers were not randomly chosen, then the Benford formula, \(P=\log _{10}\left(1+\frac{1}{d}\right),\) predicts the probability of a digit \(d\) being the first digit of the set. For example, there is a 4.6\(\%\) probability that the first digit is \(9 .\) Find the probability that the first digit is 1\(\left(\log _{10} 2 \approx 0.30103\right)\)

Solve each equation or inequality. Round to four decimal places. $$ 3^{n+2}=14.5 $$

Determine whether each function represents exponential growth or decay. $$ y=0.4\left(\frac{1}{3}\right)^{x} $$

Write each equation in exponential form. \(\log _{13} 169=2\)

Write an equivalent exponential or logarithmic equation. \(e^{-x}=5\)

Solve each equation. Check your solutions. \(\log _{3} 5+\log _{3} x=\log _{3} 10\)

Solve each equation or inequality. Round to four decimal places. $$ 9^{z-4}=6.28 $$

Determine whether each function represents exponential growth or decay. $$ y=3\left(\frac{5}{2}\right)^{x} $$

Write each equation in exponential form. \(\log _{4} \frac{1}{4}=-1\)

REVIEW A radioactive element decays over time, according to the equation $$y=x\left(\frac{1}{4}\right)^{\frac{t}{200}}$$ where \(x=\) the number of grams present initially and \(t=\) time in years. If 500 grams were present initially, how many grams will remain after 400 years?

Write an equivalent exponential or logarithmic equation. \(\ln 5.2=x\)

Solve each equation. Check your solutions. \(\log _{4} a+\log _{4} 9=\log _{4} 27\)

Solve each equation or inequality. Round to four decimal places. $$ 8.2^{n-3}=42.5 $$

Determine whether each function represents exponential growth or decay. $$ y=30^{-x} $$

Write each equation in exponential form. \(\log _{100} \frac{1}{10}=-\frac{1}{2}\)

Write an equivalent exponential or logarithmic equation. $$ e^{3}=y $$

Solve each equation. Check your solutions. \(\log _{10} 16-\log _{10}(2 t)=\log _{10} 2\)

Solve each equation or inequality. Round to four decimal places. $$ 2.1^{t-5}=9.32 $$

Determine whether each function represents exponential growth or decay. $$ y=0.2(5)^{-x} $$

Write each equation in exponential form. \(\log _{8} 4=\frac{2}{3}\)

Write an equivalent exponential or logarithmic equation. $$ e^{4 n-2}=29 $$

Solve each equation. Check your solutions. \(\log _{7} 24-\log _{7}(y+5)=\log _{7} 8\)

Solve each equation or inequality. Round to four decimal places. $$ 6^{x} \geq 42 $$

Write an exponential function for the graph that passes through the given points. $$ (0,-2) \text { and }(-2,-32) $$

Write each equation in exponential form. \(\log _{\frac{1}{5}} 25=-2\)

Write an equivalent exponential or logarithmic equation. $$ \ln 4+2 \ln x=8 $$

Solve each equation. Check your solutions. \(\log _{2} n=\frac{1}{4} \log _{2} 16+\frac{1}{2} \log _{2} 49\)

Solve each equation or inequality. Round to four decimal places. $$ 8^{2 a}<124 $$

Write an exponential function for the graph that passes through the given points. $$ (0,3) \text { and }(1,15) $$

Write each equation in logarithmic form. \(8^{3}=512\)

Solve each equation or inequality. Round to four decimal places. $$ 16^{x}=70 $$

Write an equivalent exponential or logarithmic equation. \(e^{-1}=x^{2}\)

Solve each equation. Check your solutions. \(2 \log _{10} 6-\frac{1}{3} \log _{10} 27=\log _{10} x\)

Write an exponential function for the graph that passes through the given points. $$ (0,7) \text { and }(2,63) $$