Chapter 4

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5}\left(\frac{\sqrt{x}}{25}\right) $$

With a growth rate \(k\) to double. Express each answer to the nearest whole year. Japan is growing at a rate of \(0.3 \%\) per year. How long will it take Japan to double its population?

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{4 x}-3 e^{2 x}-18=0$$

Evaluate each expression without using a calculator. $$\log _{3} 27$$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{6}\left(\frac{36}{\sqrt{x+1}}\right) $$

With a growth rate \(k\) to double. Express each answer to the nearest whole year. The logistic growth function $$f(t)=\frac{100,000}{1+5000 e^{-t}}$$ describes the number of people, \(f(t),\) who have become ill with influenza \(t\) weeks after its initial outbreak in a particular community. a. How many people became ill with the flu when the epidemic began? b. How many people were ill by the end of the fourth week? c. What is the limiting size of the population that becomes ill?

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$3^{2 x}+3^{x}-2=0$$

Evaluate each expression without using a calculator. $$\log _{7} \sqrt{7}$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=2^{x+1}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{8}\left(\frac{64}{\sqrt{x+1}}\right) $$

The logistic growth function $$ f(t)=\frac{500}{1+83.3 e^{-0.162 t}} $$ describes the population, \(f(t),\) of an endangered species of birds \(t\) years after they are introduced to a nonthreatening habitat. a. How many birds were initially introduced to the habitat? b. How many birds are expected in the habitat after 10 years? c. What is the limiting size of the bird population that the habitat will sustain?

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$2^{2 x}+2^{x}-12=0$$

Evaluate each expression without using a calculator. $$\log _{6} \sqrt{6}$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=2^{x+2}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{x^{2} y}{z^{2}}\right) $$

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. What percentage of 20 -year-olds have some coronary heart disease?

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{3} x=4$$

Evaluate each expression without using a calculator. $$\log _{2} \frac{1}{8}$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=2^{x}-1\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{x^{3} y}{z^{2}}\right) $$

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. What percentage of 80 -year-olds have some coronary heart disease?

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{5} x=3$$

Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{9}$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=2^{x}+2\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \sqrt{100 x} $$

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. At what age is the percentage of some coronary heart -disease \(50 \% ?\)

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{4}(x+5)=3$$

Evaluate each expression without using a calculator. $$\log _{64} 8$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(h(x)=2^{x+1}-1\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \sqrt{e x} $$

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. At what age is the percentage of some coronary heart disease \(70 \% ?\)

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{5}(x-7)=2$$

Evaluate each expression without using a calculator. $$\log _{81} 9$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(h(x)=2^{x+2}-1\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \sqrt[3]{\frac{x}{y}} $$

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm, and then round to three decimal places. $$ y=100(4.6)^{x} $$

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{3}(x-4)=-3$$

Evaluate each expression without using a calculator. $$\log _{5} 5$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=-2^{x}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \operatorname{og} \sqrt[5]{\frac{x}{y}} $$

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm, and then round to three decimal places. $$ y=1000(7.3)^{x} $$

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{7}(x+2)=-2$$

Evaluate each expression without using a calculator. $$\log _{11} 11$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=2^{-x}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{V x y^{3}}{z^{3}}\right) $$

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm, and then round to three decimal places. $$ y=2.5(0.7)^{x} $$

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{4}(3 x+2)=3$$

Evaluate each expression without using a calculator. $$\log _{4} 1$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=2 \cdot 2^{x}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{\sqrt[3]{x} y^{4}}{z^{5}}\right) $$