Chapter 5

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{lll}{\text { (a) } 8^{-1}=\frac{1}{8}} & {\text { (b) } 2^{-3}=\frac{1}{8}}\end{array} $$

11–14 ? Graph both functions on one set of axes. $$ f(x)=2^{x} \quad \text { and } \quad g(x)=2^{-x} $$

Find the solution of the exponential equation, correct to four decimal places. $$ 2^{3 x}=34 $$

Evaluate the expression. $$ \ln \left(\ln e^{e^{-200}}\right) $$

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } 4^{-3 / 2}=0.125} & {\text { (b) } 7^{3}=343}\end{array} $$

11–14 ? Graph both functions on one set of axes. $$ f(x)=3^{-x} \quad \text { and } \quad g(x)=\left(\frac{1}{3}\right)^{x} $$

An infectious strain of bacteria increases in number at a relative growth rate of 200% per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 10 bacteria?

Find the solution of the exponential equation, correct to four decimal places. $$ 8^{0.4 x}=5 $$

Use the Laws of Logarithms to expand the expression. $$ \log _{2}(2 x) $$

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } e^{x}=2} & {\text { (b) } e^{3}=y}\end{array} $$

The half-life of radium-226 is 1600 years. Suppose we have a 22-mg sample. (a) Find a function that models the mass remaining after \(t\) years. (b) How much of the sample will remain after 4000 years? (c) After how long will only 18 mg of the sample remain?

Find the solution of the exponential equation, correct to four decimal places. $$ 3^{x / 4}=0.1 $$

Use the Laws of Logarithms to expand the expression. $$ \log _{3}(5 y) $$

\(9-14\) Express the equation in logarithmic form. $$ \begin{array}{lll}{\text { (a) } e^{x+1}=0.5} & {\text { (b) } e^{0.5 x}=t}\end{array} $$

11–14 ? Graph both functions on one set of axes. $$ f(x)=\left(\frac{2}{3}\right)^{x} \quad \text { and } \quad g(x)=\left(\frac{4}{3}\right)^{x} $$

The half-life of cesium-137 is 30 years. Suppose we have a 10-g sample. (a) Find a function that models the mass remaining after \(t\) years. (b) How much of the sample will remain after 80 years? (c) After how long will only 2\(g\) of the sample remain?

Find the solution of the exponential equation, correct to four decimal places. $$ 5^{-x / 100}=2 $$

Use the Laws of Logarithms to expand the expression. $$ \log _{2}(x(x-1)) $$

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{3} 3} & {\text { (b) } \log _{3} 1} & {\text { (c) } \log _{3} 3^{2}}\end{array} $$

The mass \(m(t)\) remaining after \(t\) days from a 40-g sample of thorium- 234 is given by $$m(t)=40 e^{-0.0277 t}$$ (a) How much of the sample will remain after 60 days? (b) After how long will only 10 g of the sample remain? (c) Find the half-life of thorium-234.

Find the solution of the exponential equation, correct to four decimal places. $$ e^{3-5 x}=16 $$

Use the Laws of Logarithms to expand the expression. $$ \log _{5} \frac{x}{2} $$

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{5} 5^{4}} & {\text { (b) } \log _{4} 64} & {\text { (c) } \log _{9} 9}\end{array} $$

The half-life of strontium-90 is 28 years. How long will it take a 50-mg sample to decay to a mass of 32 mg?

Find the solution of the exponential equation, correct to four decimal places. $$ e^{2 x+1}=200 $$

Use the Laws of Logarithms to expand the expression. $$ \log 6^{10} $$

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{6} 36} & {\text { (b) } \log _{9} 81} & {\text { (c) } \log _{7} 7^{10}}\end{array} $$

Radium-221 has a half-life of 30 s. How long will it take for 95% of a sample to decay?

Find the solution of the exponential equation, correct to four decimal places. $$ \left(\frac{1}{4}\right)^{x}=75 $$

Use the Laws of Logarithms to expand the expression. $$ \ln \sqrt{z} $$

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{2} 32} & {\text { (b) } \log _{8} 8^{17}} & {\text { (c) } \log _{6} 1}\end{array} $$

If 250 mg of a radioactive element decays to 200 mg in 48 hours, find the half-life of the element.

Find the solution of the exponential equation, correct to four decimal places. $$ 5^{x}=4^{x+1} $$

Use the Laws of Logarithms to expand the expression. $$ \log _{2}\left(A B^{2}\right) $$

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{3}\left(\frac{1}{27}\right)} & {\text { (b) } \log _{10} \sqrt{10}} & {} & {\text { (c) } \log _{5} 0.2}\end{array} $$

After 3 days a sample of radon-222 has decayed to 58% of its original amount. (a) What is the half-life of radon-222? (b) How long will it take the sample to decay to 20% of its original amount?

Find the solution of the exponential equation, correct to four decimal places. $$ 10^{1-x}=6^{x} $$

Use the Laws of Logarithms to expand the expression. $$ \log _{6} \sqrt[4]{17} $$

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{5} 125} & {\text { (b) } \log _{49} 7} & {\text { (c) } \log _{9} \sqrt{3}}\end{array} $$

A wooden artifact from an ancient tomb contains 65% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years.)

Find the solution of the exponential equation, correct to four decimal places. $$ 2^{3 x+1}=3^{x-2} $$

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } 2^{\log _{2} 37}} & {\text { (b) } 3^{\log _{3} 8}} & {\text { (c) } e^{\ln \sqrt{5}}}\end{array} $$

The burial cloth of an Egyptian mummy is estimated to contain 59% of the carbon-14 it contained originally. How long ago was the mummy buried? (The half-life of carbon-14 is 5730 years.)

Find the solution of the exponential equation, correct to four decimal places. $$ 7^{x / 2}=5^{1-x} $$

Use the Laws of Logarithms to expand the expression. $$ \log _{2}(x y)^{10} $$

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } e^{\ln \pi}} & {\text { (b) } 10^{\log 5}} & {\text { (c) } 10^{\log 87}}\end{array} $$

A hot bowl of soup is served at a dinner party. It starts to cool according to Newton’s Law of Cooling so that its temperature at time \(t\) is given by $$T(t)=65+145 e^{-0.05 t}$$ where \(t\) is measured in minutes and \(T\) is measured in \(^{\circ} \mathrm{F}\) (a) What is the initial temperature of the soup? (b) What is the temperature after 10 \(\mathrm{min}\) ? (c) After how long will the temperature be \(100^{\circ} \mathrm{F} ?\)

Find the solution of the exponential equation, correct to four decimal places. $$ \frac{50}{1+e^{-x}}=4 $$

Use the Laws of Logarithms to expand the expression. $$ \log _{5} \sqrt[3]{x^{2}+1} $$

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{8} 0.25} & {\text { (b) } \ln e^{4}} & {\text { (c) } \ln (1 / e)} & {}\end{array} $$