Chapter 5

Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ h(x)=(1.1)^{x} $$

Bacteria Culture A culture starts with 8600 bacteria. After one hour the count is \(10,000\) . (a) Find a function that models the number of bacteria \(n(t)\) after \(t\) hours. (b) Find the number of bacteria after 2 hours. (c) After how many hours will the number of bacteria double?

Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } 5^{3}=125} & {\text { (b) } 10^{-4}=0.0001}\end{array} $$

Find the solution of the exponential equation, rounded to four decimal places. \(4+3^{5 x}=8\)

\(7-18\) Evaluate the expression. $$ \log _{2} 6-\log _{2} 15+\log _{2} 20 $$

Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ g(x)=3(1.3)^{x} $$

Bacteria Culture The count in a culture of bacteria was 400 after 2 hours and \(25,600\) after 6 hours. (a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage. (b) What was the initial size of the culture? (c) Find a function that models the culture? (d) Find the number of bacteria after 4.5 hours. (e) When will the number of bacteria be \(50,000 ?\)

Express the equation in logarithmic form. $$ \begin{array}{ll}{\text { (a) } 10^{3}=1000} & {\text { (b) } 81^{1 / 2}=9}\end{array} $$

Find the solution of the exponential equation, rounded to four decimal places. \(2^{3 x}=34\)

\(7-18\) Evaluate the expression. $$ \log _{3} 100-\log _{3} 18-\log _{3} 50 $$

Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ h(x)=2\left(\frac{1}{4}\right)^{x} $$

Population of California The population of California was 29.76 million in 1990 and 33.87 million in 2000 . Assume that the population grows exponentially. (a) Find a function that models the population \(t\) years after 1990 . (b) Find the time required for the population to double. (c) Use the function from part (a) to predict the popuble. California in the year 2010 . Look up California's actual population in 2010 , and compare.

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

Find the solution of the exponential equation, rounded to four decimal places. \(8^{0.4 x}=5\)

\(7-18\) Evaluate the expression. $$ \log _{4} 16^{100} $$

The hyperbolic cosine function is defined by $$ \cosh (x)=\frac{e^{x}+e^{-x}}{2} $$ (a) Sketch the graphs of the functions \(y=\frac{1}{2} e^{x}\) and \(y=\frac{1}{2} e^{-x}\) on the same axes, and use graphical addition (see Section 3.6\()\) to sketch the graph of \(y=\cosh (x)\) . (b) Use the definition to show that \(\cosh (-x)=\cosh (x)\)

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

World Population The population of the world was 5.7 billion in \(1995,\) and the observed relative growth rate was 2\(\%\) per year. (a) By what year will the population have doubled? (b) By what year will the population have tripled?

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

Find the solution of the exponential equation, rounded to four decimal places. \(e^{3-5 x}=16\)

\(7-18\) Evaluate the expression. $$ \log _{2} 8^{33} $$

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

\(17-24\) . These exercises use the radioactive decay model. Radioactive Radium The half-life of radium- 226 is 1600 years. Suppose we have a 22 -mg sample. (a) Find a function \(m(t)=m_{0} 2^{-4 / h}\) that models the mass remaining after \(t\) years. (b) Find a function \(m(t)=m_{0} e^{-r t}\) that models the mass remaining after \(t\) years. (c) How much of the sample will remain after 4000 years? (d) After how long will only 18 mg of the sample remain?

Express the equation in logarithmic form. $$ e^{x}=2 \quad \text { (b) } e^{3}=y $$

Find the solution of the exponential equation, rounded to four decimal places. \(5^{-x / 100}=2\)

\(7-18\) Evaluate the expression. $$ \log \left(\log 10^{10,000}\right) $$

(a) Draw the graphs of the family of functions $$ f(x)=\frac{a}{2}\left(e^{x / a}+e^{-x / a}\right) $$ for \(a=0.5,1,1.5,\) and 2 (b) How does a larger value of \(a\) affect the graph?

Graph both functions on one set of axes. $$ f(x)=4^{x} \quad \text { and } \quad g(x)=7^{x} $$

Express the equation in logarithmic form. $$ e^{x+1}=0.5 \quad \text { (b) } e^{0.5 x}=t $$

Find the solution of the exponential equation, rounded to four decimal places. \(e^{3-5 x}=16\)

\(7-18\) Evaluate the expression. $$ \ln \left(\ln e^{e^{200}}\right) $$

18-19 \(\mathbf{m}\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ g(x)=x^{x} \quad(x>0) $$

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

\(17-24\) . These exercises use the radioactive decay model. Radioactive Strontium 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 \(\mathrm{mg}\) ?

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

Find the solution of the exponential equation, rounded to four decimal places. \(e^{2 x+1}=200\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{2}(2 x) $$

18-19 \(\mathbf{m}\) Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$ g(x)=e^{x}+e^{-3 x} $$

\(17-24\) . These exercises use the radioactive decay model. Radioactive Radium Radium- 221 has a half-life of 30 \(\mathrm{s}\) . How long will it take for 95\(\%\) of a sample to decay?

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{1} 5^{4}} & {\text { (b) } \log _{4} 64} & {\text { (c) } \log _{1} 9}\end{array} $$

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

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{3}(5 y) $$

\(17-24\) . These exercises use the radioactive decay model. Finding Half-life If 250 \(\mathrm{mg}\) of a radioactive element de- cays to 200 \(\mathrm{mg}\) in 48 hours, find the half-life of the element.

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{4} 36} & {\text { (b) } \log _{3} 81} & {\text { (c) } \log _{7} 7^{10}}\end{array} $$

Find the solution of the exponential equation, rounded to four decimal places. \(5^{x}=4^{x+1}\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{2}(x(x-1)) $$

Radioactive Decay A radioactive substance decays in such a way that the amount of mass remaining after \(t\) days is given by the function $$ m(t)=13 e^{-0.015 t} $$ where \(m(t)\) is measured in kilograms. (a) Find the mass at time \(t=0\) . (b) How much of the mass remains after 45 days?

\(17-24\) . These exercises use the radioactive decay model. Radioactive Radon 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?

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{2} 32} & {\text { (b) } \log _{x} 8^{17}} & {\text { (c) } \log _{4} 1}\end{array} $$

Find the solution of the exponential equation, rounded to four decimal places. \(10^{1-x}=6^{x}\)