Chapter 4

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

Evaluate the expression. $$\log _{12} 9+\log _{12} 16$$

Express the equation in logarithmic form. (a) \(5^{3}=125\) (b) \(10^{-4}=0.0001\)

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

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

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

Express the equation in logarithmic form. (a) \(10^{3}=1000\) (b) \(81^{1 / 2}=9\)

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

These exercises use the population growth model. 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 number of bacteria \(n(t)\) after \(t\) hours. (d) Find the number of bacteria after 4.5 hours. (e) When will the number of bacteria be \(50,000 ?\)

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

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

Graph the function, not by plotting points, but by starting from the graph of \(y=e^{x}\) in Figure \(1 .\) State the domain, range, and asymptote. $$g(x)=-e^{x-1}-2$$

Express the equation in logarithmic form. (a) \(8^{-1}=\frac{1}{8}\) (b) \(2^{-3}=\frac{1}{8}\)

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

These exercises use the population growth model. 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 population of California in the year \(2010 .\) Look up California's actual population in \(2010,\) and compare.

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

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 2.6 ) to sketch the graph of \(y=\cosh (x).\) (b) Use the definition to show that \(\cosh (-x)=\cosh (x).\)

Express the equation in logarithmic form. (a) \(4^{-3 / 2}=0.125\) (b) \(7^{3}=343\)

Find the solution of the exponential equation, rounded to four decimal places. $$3^{x / 14}=0.1$$

These exercises use the population growth model. 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?

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

Evaluate the expression. $$\log _{2} 8^{33}$$

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

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

These exercises use the radioactive decay model. The half-life of radium- 226 is 1600 years. Suppose we have a 22 -mg sample. (a) Find a function \(m(t)=m_{0} 2^{-t / h}\) 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?

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

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

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

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

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

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

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. $$g(x)=x^{x} \quad(x>0)$$

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

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

Evaluate the expression. (a) \(\log _{3} 3\) (b) \(\log _{3} 1\) (c) \(\log _{3} 3^{2}\)

These exercises use the radioactive decay model. 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?

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

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. $$g(x)=e^{x}+e^{-3 x}$$

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

Evaluate the expression. (a) \(\log _{5} 5^{4}\) (b) \(\log _{4} 64\) (c) \(\log _{3} 9\)

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

Medical Drugs When a certain medical drug is administered to a patient, the number of milligrams remaining in the patient's bloodstream after \(t\) hours is modeled by $$D(t)=50 e^{-0.2 t}$$ How many milligrams of the drug remain in the patient's bloodstream after 3 hours?

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

Evaluate the expression. (a) \(\log _{6} 36\) (b) \(\log _{9} 81\) (c) \(\log _{7} 7^{10}\)

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

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

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?

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

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