Chapter 9

Evaluate each expression. \(\log _{4} 16^{x}\)

Solve each equation. Check your solution. $$ \left(\frac{1}{7}\right)^{y-3}=343 $$

If you deposit \(\$ 100\) in an account paying 3.5\(\%\) interest compounded continuously, how long will it take for your money to double?

Solve each equation. Check your solution. $$ 10^{x-1}=100^{2 x-3} $$

Suppose you deposit A dollars in an account paying an interest rate of r, compounded continuously. Write an equation giving the time t needed for your money to double, or the doubling time.

STAR LIGHT For Exercises \(42-44,\) use the following information. The brightness, or apparent magnitude, \(m\) of a star or planet is given by \(m=6-2.5 \log _{10} \frac{L}{L_{0}},\) where \(L\) is the amount of light \(L\) coming to Earth from the star or planet and \(L_{0}\) is the amount of light from a sixth magnitude star. RESEARCH Use the Internet or other reference to find the magnitude of the dimmest stars that we can now see with ground-based telescopes.

Solve each equation. Round to four decimal places. $$ 20^{x^{2}}=70 $$

Solve each equation. Check your solutions. \(\log _{9} x=2\)

Solve each equation. Check your solution. $$ 36^{2 p}=216^{p-1} $$

Solve each equation. Round to four decimal places. $$ 2^{x^{2}-3}=15 $$

Solve each equation. Check your solutions. \(\log _{25} n=\frac{3}{2}\)

Solve each inequality. Check your solution. $$ 3^{n-2}>27 $$

REASONING Use the properties of Logarithms to prove that \(\log _{a} \frac{1}{x}=-\log _{a} x\)

Solve each equation. Round to four decimal places. $$ 2^{2 x+3}=3^{3 x} $$

Solve each equation. Check your solutions. \(\log _{\frac{1}{7}} x=-1\)

Solve each inequality. Check your solution. $$ 2^{2 n} \leq \frac{1}{16} $$

Solve each equation or inequality. Round to the nearest ten-thousandth. \(\ln 2 x=4\)

Solve each equation. Round to four decimal places. $$ 16^{d-4}=3^{3-d} $$

Solve each equation. Check your solutions. \(\log _{10}\left(x^{2}+1\right)=1\)

Solve each inequality. Check your solution. $$ 16^{n}<8^{n+1} $$

Solve each equation or inequality. Round to the nearest ten-thousandth. \(\ln 3 x=5\)

CHALLENGE Simplify \(x^{3 \log _{x} 2-\log _{x} 5}\) to find an exact numerical value.

Solve each equation. Round to four decimal places. $$ 5^{5 y-2}=2^{2 y+1} $$

Solve each equation. Check your solutions. \(\log _{b} 64=3\)

Solve each inequality. Check your solution. $$ 32^{5 p+2} \geq 16^{5 p} $$

Solve each equation or inequality. Round to the nearest ten-thousandth. \(\ln (x+1)=1\)

CHALLENGE Tell whether each statement is true or false. If true, show that it is true. If false, give a counterexample. For all positive numbers \(m, n,\) and \(b,\) where \(b \neq 1, \log _{b}(m+n)=\) \(\log _{b} m+\log _{b} n\)

Solve each equation. Round to four decimal places. $$ 8^{2 x-5}=5^{x+1} $$

Solve each equation. Check your solutions. \(\log _{b} 121=2\)

Sketch the graph of each function. Then state the function's domain and range. $$ y=-\left(\frac{1}{5}\right)^{x} $$

Solve each equation or inequality. Round to the nearest ten-thousandth. \(\ln (x-7)=2\)

CHALLENGE Tell whether each statement is true or false. If true, show that it is true. If false, give a counterexample. For all positive numbers \(m, n, x,\) and \(b,\) where \(b \neq 1, n \log _{b} x+m \log _{b} x=\) \((n+m) \log _{b} x\)

Solve each equation. Round to four decimal places. $$ 2^{n}=\sqrt{3^{n-2}} $$

Sketch the graph of each function. Then state the function's domain and range. $$ y=-2.5(5)^{x} $$

Solve each equation or inequality. Round to the nearest ten-thousandth. \(e^{x}<4.5\)

REASONING Use the properties of exponents to prove the Quotient Property of Logarithms.

Solve each equation. Round to four decimal places. $$ 4^{x}=\sqrt{5^{x+2}} $$

Solve each equation or inequality. Round to the nearest ten-thousandth. \(e^{x}>1.6\)

Solve each equation. Round to four decimal places. $$ 3^{y}=\sqrt{2^{y-1}} $$

Solve each equation or inequality. Check your solutions. \(\log _{2} c>8\)

Solve each equation or inequality. Round to the nearest ten-thousandth. \(e^{5 x} \geq 25\)

\(\mathrm{ACT} / \mathrm{SAT}\) To what is \(2 \log _{5} 12-\log _{5} 8-\) 2 \(\log _{5} 3\) equal? \(\mathrm{A} \log _{5} 2\) \(\mathrm{B} \log _{5} 3\) \(\mathrm{C} \log _{5} 0.5\) \(\mathrm{D} 1\)

For Exercises 53 and \(54,\) use the following information. The musical cent is a unit in a logarithmic scale of relative pitch or intervals. One octave is equal to 1200 cents. The formula to determine the difference in cents between two notes with frequencies \(a\) and \(b\) is \(n=1200\left(\log _{2} \frac{a}{b}\right)\). Find the interval in cents when the frequency changes from 443 Hertz \((\mathrm{Hz})\) to 415 \(\mathrm{Hz}\) .

Solve each equation or inequality. Check your solutions. \(\log _{64} y \leq \frac{1}{2}\)

For Exercises \(53-55,\) use the following information. Every ten years, the Bureau of the Census counts the number of people living in the United States. In \(1790,\) the population of the U.S. was 3.93 million. By \(1800,\) this number had grown to 5.31 million. Write an exponential function that could be used to model the U.S. population \(y\) in millions for 1790 to \(1800 .\) Write the equation in terms of \(x,\) the number of decades \(x\) since \(1790 .\)

Solve each equation or inequality. Round to the nearest ten-thousandth. \(e^{-2 x} \leq 7\)

REVIEW In a movie theater, 2 boys and 3 girls are seated randomly together. What is the probability that the 2 boys are seated next to each othor? $$ F \frac{1}{5} \quad \text { G } \frac{2}{5} \quad H \frac{1}{2} \quad J \frac{2}{3} $$

For Exercises 53 and \(54,\) use the following information. The musical cent is a unit in a logarithmic scale of relative pitch or intervals. One octave is equal to 1200 cents. The formula to determine the difference in cents between two notes with frequencies \(a\) and \(b\) is \(n=1200\left(\log _{2} \frac{a}{b}\right)\). If the interval is 55 cents and the beginning frequency is 225 \(\mathrm{Hz}\) , find the final frequency.

Solve each equation or inequality. Check your solutions. \(\log _{\frac{1}{3}} p<0\)

For Exercises \(53-55,\) use the following information. Every ten years, the Bureau of the Census counts the number of people living in the United States. In \(1790,\) the population of the U.S. was 3.93 million. By \(1800,\) this number had grown to 5.31 million. Assume that the U.S. population continued to grow at least that fast. Estimate the population for the years \(1820,1840,\) and \(1860 .\) Then compare your estimates with the actual population for those years, which were 9.64 \(17.06,\) and 31.44 million, respectively.