Chapter 4

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$y=2^{-x^{2}}$$

The IQ scores for adults roughly follow the normal distribution \(y=0.0266 e^{-(x-100)^{2 / 450}}\) \(70 \leq x \leq 115,\) where \(x\) is the IQ score. (a) Use a graphing utility to graph the function. (b) Use the graph in part (a) to estimate the average IQ score.

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\ln \frac{6}{e^{2}}$$.

Sketch the graph of \(f .\) Then use the graph of \(f\) to sketch the graph of \(g.\) $$\begin{aligned}&f(x)=15^{x}\\\&g(x)=\log _{15} x\end{aligned}$$

Solve the logarithmic equation. $$\log _{11} x=-1$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$y=3^{x-2}+1$$

Find the slope and \(y\)-intercept of the equation of the line. Then sketch the line by hand. $$2 x+5 y=10$$

The sales \(S\) (in thousands of units) of a cleaning solution after \(x\) hundred dollars is spent on advertising are given by \(S=10\left(1-e^{k x}\right) .\) When \(\$ 500\) is spent on advertising, 2500 units are sold. (a) Complete the model by solving for \(k\) (b) Estimate the number of units that will be sold when advertising expenditures are raised to \(\$ 700 .\)

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\ln \frac{e^{5}}{7}$$.

Sketch the graph of \(f .\) Then use the graph of \(f\) to sketch the graph of \(g.\) $$\begin{aligned}&f(x)=4^{x}\\\&g(x)=\log _{4} x\end{aligned}$$

Solve the logarithmic equation. $$\log _{10} x=-\frac{1}{4}$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$y=4^{x+1}-2$$

Find the slope and \(y\)-intercept of the equation of the line. Then sketch the line by hand. $$3 x-2 y=9$$

A conservation organization releases 100 animals of an endangered species into a game preserve. The organization believes that the preserve has a carrying capacity of 1000 animals and that the growth of the herd will follow the logistic curve $$p(t)=\frac{1000}{1+9 e^{-0.1656 t}}$$ where \(t\) is measured in months. (a) What is the population after 5 months? (b) After how many months will the population reach \(500 ?\) (c) Use a graphing utility to graph the function. Use the graph to determine the values of \(p\) at which the horizontal asymptotes occur. Identify the asymptote that is most relevant in the context of the problem and interpret its meaning.

Use the properties of logarithms to verify the equation.$$\log _{5} \frac{1}{250}=-3-\log _{5} 2$$.

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. $$y=\log _{10}(x+2)$$

Solve the logarithmic equation. $$\ln (2 x-1)=5$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=e^{-x}$$

Find the slope and \(y\)-intercept of the equation of the line. Then sketch the line by hand. $$0.4 x-2.5 y=12.5$$

The number \(Y\) of yeast organisms in a culture is given by the model $$Y=\frac{663}{1+72 e^{-0.547 t}}$$ where \(t\) represents the time (in hours). (a) Use a graphing utility to graph the model. (b) Use the model to predict the populations for the 19th hour and the 30 th hour. (c) According to this model, what is the limiting value of the population? (d) Why do you think this population of yeast follows a logistic growth model instead of an exponential growth model?

Use the properties of logarithms to verify the equation.$$-\ln 24=-(3 \ln 2+\ln 3)$$.

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. $$y=\log _{10}(x-1)$$

Solve the logarithmic equation. $$\ln (3 x+5)=8$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$s(t)=3 e^{-0.2 t}$$

Find the slope and \(y\)-intercept of the equation of the line. Then sketch the line by hand. $$1.2 x+3.5 y=10.5$$

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{10} 10 x$$.

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. $$y=1+\log _{10} x$$

Simplify the expression. $$\ln e^{x^{2}}$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=3 e^{x+4}$$

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{10} 100 x$$

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. $$y=2-\log _{10} x$$

Simplify the expression. $$\ln e^{2 x-1}$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=2 e^{x-3}$$

Use the following information for determining sound intensity. The level of sound \(\beta\) (in decibels) with an intensity \(I\) is $$\beta=10 \log _{10} \frac{I}{I_{0}}$$ where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 49 and \(50,\) find the Ievel of each sound \(\beta\). (a) \(I=10^{-10}\) watt per \(m^{2}\) (quiet room) (b) \(I=10^{-5}\) watt per \(\mathrm{m}^{2}\) (busy street corner) (c) \(I \approx 10^{0}\) watt per \(\mathrm{m}^{2}\) (threshold of pain)

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{10} \frac{t}{8}$$.

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. $$y=1+\log _{10}(x-2)$$

Simplify the expression. $$e^{\ln \left(x^{2}-3\right)}$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=2+e^{x-5}$$

Use the following information for determining sound intensity. The level of sound \(\beta\) (in decibels) with an intensity \(I\) is $$\beta=10 \log _{10} \frac{I}{I_{0}}$$ where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 49 and \(50,\) find the Ievel of each sound \(\beta\). (a) \(I=10^{-4}\) watt per \(m^{2}\) (door slamming) (b) \(I=10^{-3}\) watt per \(m^{2}\) (loud car horn) (c) \(I=10^{-2}\) watt per \(m^{2}\) (siren at 30 meters)

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{10} \frac{7}{z}$$

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. $$y=2+\log _{10}(x+1)$$

Simplify the expression. $$e^{\ln x^{2}}$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$g(x)=e^{x+1}+2$$

Use the following information for determining sound intensity. The level of sound \(\beta\) (in decibels) with an intensity \(I\) is $$\beta=10 \log _{10} \frac{I}{I_{0}}$$ where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 49 and \(50,\) find the Ievel of each sound \(\beta\). As a result of the installation of a muffler, the noise level of an engine was reduced from 88 to 72 decibels. Find the percent decrease in the intensity level of the noise due to the installation of the muffler.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{8} x^{4}$$.

Simplify the expression. $$-1+\ln e^{2 x}$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$s(t)=2 e^{-0.12 t}$$

Use the following information for determining sound intensity. The level of sound \(\beta\) (in decibels) with an intensity \(I\) is $$\beta=10 \log _{10} \frac{I}{I_{0}}$$ where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 49 and \(50,\) find the Ievel of each sound \(\beta\). As a result of the installation of noise suppression materials, the noise level in an auditorium was reduced from 93 to 80 decibels. Find the percent decrease in the intensity level of the noise due to the installation of these materials.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{6} z^{-3}$$.

Simplify the expression. $$-4+e^{\ln x^{4}}$$