Chapter 9

The altimeter in an airplane gives the altitude or height \(h(\text { in feet })\) of a plane above sea level by measuring the outside air pressure \(P\) (in kilopascals). The height and air pressure are related by the model \(P=101.3 e^{-\frac{h}{26,200}}\). Find a formula for the height in terms of the outside air pressure.

Solve each equation. Check your solutions. \(\log _{10} a+\log _{10}(a+21)=2\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{7} 5 $$

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

MONEY For Exercises 10 and 11 , use the following information. In \(1993,\) My- Lien inherited \(\$ 1,000,000\) from her grandmother. She invested all of the money, and by 2005 , the amount had grown to \(\$ 1,678,000\) . Assume that the amount of money continues to grow at the same rate. Estimate the amount of money in \(2015 .\) Is this estimate reasonable? Explain your reasoning.

The Martins bought a condominium for \(\$ 145,000 .\) Assuming that the value of the condo will appreciate at most 5\(\%\) a year, how much will the condo be worth in 5 years?

The altimeter in an airplane gives the altitude or height \(h(\text { in feet })\) of a plane above sea level by measuring the outside air pressure \(P\) (in kilopascals). The height and air pressure are related by the model \(P=101.3 e^{-\frac{h}{26,200}}\). Use the formula you found in Exercise 11 to approximate the height of a plane above sea level when the outside air pressure is 57 kilopascals.

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} 50\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{3} 42 $$

Solve each equation. Check your solution. $$ 2^{n+4}=\frac{1}{32} $$

For Exercises 13 and \(14,\) use the following information. The annual Gross Domestic Product (GDP) of a country is the value of all of the goods and services produced in the country during a year. During the period \(2001-2001-2004,\) the Gross Domestic Product of the United States grew about \(2.8 \% 1-2001\) per year, measured in 2004 dollars. In \(2001,\) the GDP was \(\$ 9891\) billion. Assuming this rate of growth continues, what will the GDP of the United States be in the year 2015\(?\)

Solve each equation or inequality. Round to the nearest ten-thousandth. \(e^{x}>30\)

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} 30\)

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{2} 9 $$

An equation for loudness \(L\) in decibels, is \(L=10 \log _{10} R\) where \(R\) is the relative intensity of the sound. Solve \(130=10 \log _{10} R\) to find the relative intensity of a fireworks display with a loudness of 130 decibels.

Solve each equation. Check your solution. $$ 9^{2 y-1}=27^{y} $$

Solve each equation or inequality. Round to the nearest ten-thousandth. \(\ln x<6\)

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} 20\)

Use a calculator to evaluate each expression to four decimal places. $$ \log 5 $$

Solve each equation. Check your solution. $$ 4^{3 x+2}=\frac{1}{256} $$

An equation for loudness \(L\) in decibels, is \(L=10 \log _{10} R\) where \(R\) is the relative intensity of the sound. Solve \(75=10 \log _{10} R\) to find the relative intensity of a concert with a loudness of 75 decibels.

Solve each equation or inequality. Round to the nearest ten-thousandth. \(2 \ln 3 x+1=5\)

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} \frac{2}{3}\)

Use a calculator to evaluate each expression to four decimal places. $$ \log 12 $$

Solve each inequality. Check your solution. $$ 5^{2 x+3} \leq 125 $$

For Exercises 15 and \(16,\) use the following information. Bacteria usually reproduce by a process known as binary fission. In this type of reproduction, one bacterium divides, forming two bacteria. Under ideal conditions, some bacteria reproduce every 20 minutes. Write the equation for modeling the exponential growth of this bacterium.

Solve each equation or inequality. Round to the nearest ten-thousandth. \(\ln x^{2}=9\)

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} \frac{3}{2}\)

Use a calculator to evaluate each expression to four decimal places. $$ \log 7.2 $$

Solve each inequality. Check your solutions. \(\log _{4} x<2\)

Solve each inequality. Check your solution. $$ 3^{3 x-2}>81 $$

In \(1928,\) when the high jump was first introduced as a women's sport at the Olympic Games, the winning women's jump was 62.5 inches, while the winning men's jump was 76.5 inches. Since then, the winning jump for women has increased by about 0.38\(\%\) per year, while the winning jump for men has increased at a slower rate, 0.3\(\%\) . If these rates continue, when will the women's winning high jump be higher than the men's?

Use a calculator to evaluate each expression to four decimal places. \(e^{4}\)

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} \frac{4}{3}\)

Solve each inequality. Check your solutions. \(\log _{3}(2 x-1) \leq 2\)

Solve each inequality. Check your solution. $$ 4^{4 a+6} \leq 16^{a} $$

The Mendes family bought a new house 10 years ago for \(\$ 120,000 .\) The house is now worth \(\$ 191,000 .\) Assuming a steady rate of growth, what was the yearly rate of appreciation?

Use a calculator to evaluate each expression to four decimal places. \(e^{5}\)

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} 9\)

Sketch the graph of each function. Then state the function's domain and range. $$ y=2(3)^{x} $$

Solve each inequality. Check your solutions. \(\log _{16} x \geq \frac{1}{4}\)

Use a calculator to evaluate each expression to four decimal places. \(e^{-1.2}\)

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} 8\)

Sketch the graph of each function. Then state the function's domain and range. $$ y=5(2)^{x} $$

Solve each equation. Check your solutions. \(\log _{5}(3 x-1)=\log _{5}\left(2 x^{2}\right)\)

Use a calculator to evaluate each expression to four decimal places. \(e^{0.5}\)

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} 16\)

The acidity of water determines the toxic effects of runoff into streams from industrial or agricultural areas. A pH range of 6.0 to 9.0 appears to provide protection for freshwater fish. What is this range in terms of the water's hydrogen ion concentration?

Sketch the graph of each function. Then state the function's domain and range. $$ y=0.5(4)^{x} $$

Use a calculator to evaluate each expression to four decimal places. \(\ln 3\)