Chapter 10

Explain how you would solve the equation $$ \left(2^{x+1}\right)\left(8^{2 x-3}\right)=64 $$

What rate of interest compounded continuously is needed for an investment of \(\$ 2500\) to grow to \(\$ 10,000\) in 20 years? \(6.9 \%\)

Graph each of the functions. Remember that the graph of \(f(x)=\log _{2} x\) is given in Figure 10.27. $$ f(x)=-2+\log _{2} x $$

Solve each equation. \(\log _{9} x=-\frac{5}{2}\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=x^{2}+1 \quad \text { for } x \leq 0 $$

Why is the base of an exponential function restricted to positive numbers not including 1 ?

For a certain strain of bacteria, the number of bacteria present after \(t\) hours is given by the equation \(Q=\) \(Q_{0} e^{0.34 t}\), where \(Q_{0}\) represents the initial number of bacteria. How long will it take 400 bacteria to increase to 4000 bacteria?

Solve each equation. \(\log _{x} 2=\frac{1}{2}\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=1+\frac{1}{x} \quad \text { for } x>0 $$

Complete the following chart, which illustrates what happens to \(\$ 1000\) invested at various rates of interest for different lengths of time but always compounded continuously. Round your answers to the nearest dollar. $$ \begin{aligned} &\$ 1000 \text { Compounded continuously See answer section }\\\ &\begin{array}{l|c|c|c|c} \hline & 8 \% & 10 \% & 12 \% & 14 \% \\ \hline \begin{array}{c} \text { 5 years } \\ 10 \text { years } \\ \text { 15 years } \end{array} & & & & \\ 20 \text { years } & & & & \\ 25 \text { years } & & & & \\ \hline \end{array} \end{aligned} $$

Explain how you would graph the function $$ f(x)=-\left(\frac{1}{3}\right)^{x} $$

A piece of machinery valued at \(\$ 30,000\) depreciates at a rate of \(10 \%\) yearly. How long will it take for it to reach a value of \(\$ 15,000\) ?

Solve each equation. \(\log _{x} 3=\frac{1}{2}\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=\frac{x}{x+1} \quad \text { for } x>-1 $$

The equation \(P(a)=14.7 e^{-0.21 a}\), where \(a\) is the altitude above sea level measured in miles, yields the atmospheric pressure in pounds per square inch. If the atmospheric pressure at Cheyenne, Wyoming, is approximately \(11.53\) pounds per square inch, find that city's altitude above sea level. Express your answer to the nearest hundred feet. 6100 feet

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2} 35\)

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=3 x $$

Graph \(f(x)=2^{x}\). Where should the graphs of \(f(x)=\) \(2^{x-5}, f(x)=2^{x-7}\), and \(f(x)=2^{x+5}\) be located? Graph all three functions on the same set of axes with \(f(x)=2^{x}\).

The number of grams of a certain radioactive substance present after \(t\) hours is given by the equation \(Q=\) \(Q_{0} e^{-0.45 t}\), where \(Q_{0}\) represents the initial number of grams. How long will it take 2500 grams to be reduced to 1250 grams?

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2}\left(\frac{7}{5}\right)\)

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=-x $$

For Problems 52-56, graph each of the functions. $$ f(x)=x\left(2^{x}\right) $$

Graph \(f(x)=3^{x}\). Where should the graphs of \(f(x)=\) \(3^{x}+2, f(x)=3^{x}-3\), and \(f(x)=3^{x}-7\) be located? Graph all three functions on the same set of axes with \(f(x)=3^{x}\).

For a certain culture, the equation \(Q(t)=Q_{0} e^{0.4 t}\), where \(Q_{0}\) is an initial number of bacteria and \(t\) is time measured in hours, yields the number of bacteria as a function of time. How long will it take 500 bacteria to increase to 2000 ?

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=2 x+1 $$

Graph each of the functions. $$ f(x)=\frac{e^{x}+e^{-x}}{2} $$

Graph \(f(x)=\left(\frac{1}{2}\right)^{x}\). Where should the graphs of \(f(x)=-\left(\frac{1}{2}\right)^{x}, f(x)=\left(\frac{1}{2}\right)^{-x}\), and \(f(x)=-\left(\frac{1}{2}\right)^{-x}\) be located? Graph all three functions on the same set of axes with \(f(x)=\left(\frac{1}{2}\right)^{x}\).

Suppose that the equation \(P(t)=P_{0} e^{0.02 t}\), where \(P_{0}\) represents an initial population and \(t\) is the time in years, is used to predict population growth. How long will it take a city of 50,000 to double its population?

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\log 7}{\log 3} $$

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2} 49\)

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=-3 x-3 $$

Graph each of the functions. $$ f(x)=\frac{2}{e^{x}+e^{-x}} $$

An earthquake in Los Angeles in 1971 had an intensity of approximately 5 million times the reference intensity. What was the Richter number associated with that earthquake?

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\ln 2}{\ln 7} $$

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2} \sqrt{7}\)

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=\frac{2}{x-1} \quad \text { for } x>1 $$

Graph each of the functions. $$ f(x)=\frac{e^{x}-e^{-x}}{2} $$

What is the solution for \(3^{x}=5\) ? Do you agree that it is between 1 and 2 because \(3^{1}=3\) and \(3^{2}=9\) ? Now graph \(f(x)=3^{x}-5\) and use the ZOOM and TRACE features of your graphing calculator to find an approximation, to the nearest hundredth, for the \(x\) intercept. You should get an answer of 1.46. Do you see that this is an approximation for the solution of \(3^{x}=5\) ? Try it; raise 3 to the \(1.46\) power. Find an approximate solution, to the nearest hundredth, for each of the following equations by graphing the appropriate function and finding the \(x\) intercept. (a) \(2^{x}=19\) (b) \(3^{x}=50\) (c) \(4^{x}=47\) (d) \(5^{x}=120\) (e) \(2^{x}=1500\) (f) \(3^{x-1}=34\)

An earthquake in San Francisco in 1906 was reported to have a Richter number of \(8.3\). How did its intensity compare to the reference intensity? 200 million times more than the reference

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{2 \ln 3}{\ln 8} $$

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2} \sqrt[3]{5}\)

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=\frac{-1}{x-2} \quad \text { for } x>2 $$

Graph each of the functions. $$ f(x)=\frac{2}{e^{x}-e^{-x}} $$

Calculate how many times more intense an earthquake with a Richter number of \(7.3\) is than an earthquake with a Richter number of 6.4. Approximately 8 times

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\ln 5}{2 \ln 3} $$

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2} 175\)

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=x^{2}-4 \quad \text { for } x \geq 0 $$

Perform the following calculations and express answers to the nearest hundredth. $$ \frac{\ln 3}{0.04} $$

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2} 56\)

(a) find \(f^{-1}\) and (b) graph \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=\sqrt{x-3} \text { for } x \geq 3 $$