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. $$g(x)=e^{0.5 x}-1$$

Use the acidity model \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\) where acidity (pH) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (in moles of hydrogen per liter) of a solution. Find the \(\mathrm{pH}\) when \(\left[\mathrm{H}^{+}\right]=2.3 \times 10^{-5}\)

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

Simplify the expression. $$5-e^{\ln \left(x^{2}+1\right)}$$

Use a graphing utility to (a) graph the function and (b) find any asymptotes numerically by creating a table of values for the function. $$f(x)=\frac{8}{1+e^{-0.5 x}}$$

Use the acidity model \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\) where acidity (pH) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (in moles of hydrogen per liter) of a solution. Compute \(\left[\mathrm{H}^{+}\right]\) for a solution for which \(\mathrm{pH}=5.8\)

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\ln \sqrt[3]{t}$$.

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

Use a graphing utility to (a) graph the function and (b) find any asymptotes numerically by creating a table of values for the function. $$g(x)=\frac{8}{1+e^{-0.5 / x}}$$

Use the acidity model \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\) where acidity (pH) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (in moles of hydrogen per liter) of a solution. A grape has a pH of \(3.5,\) and baking soda has a pH of \(8.0 .\) The hydrogen ion concentration of the grape is how many times that of the baking soda?

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\ln x y z$$.

Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{10} x, \quad g(x)=\log _{10}(-x)$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$8^{3 x}=360$$

Use a graphing utility to (a) graph the function and (b) find any asymptotes numerically by creating a table of values for the function. $$f(x)=-\frac{6}{2-e^{0.2 x}}$$

Use the acidity model \(\mathbf{p H}=-\log \left[\mathbf{H}^{+}\right]\) where acidity (pH) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (in moles of hydrogen per liter) of a solution. The \(\mathrm{pH}\) of a solution is decreased by one unit. The hydrogen ion concentration is increased by what factor?

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

Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{10} x, \quad g(x)=\log _{10}(x+7)$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$6^{5 x}=3000$$

Use a graphing utility to (a) graph the function and (b) find any asymptotes numerically by creating a table of values for the function. $$f(x)=\frac{6}{2-e^{0.2 / x}}$$

The total interest \(u\) paid on a home mortgage of \(P\) dollars at interest rate \(r\) for \(t\) years is given by $$u=P\left[\frac{r t}{1-\left(\frac{1}{1+r / 12}\right)^{12 t}}-1\right]$$ Consider a \(\$ 230,000\) home mortgage at \(3 \%\) (a) Use a graphing utility to graph the total interest function. (b) Approximate the length of the mortgage when the total interest paid is the same as the amount of the mortgage. Is it possible that a person could pay twice as much in interest charges as the amount of the mortgage?

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{6} a b^{3} c^{-2}$$.

Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{2} x, \quad g(x)=4-\log _{2} x$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$5^{-t / 2}=0.20$$

Use a graphing utility to find the point(s) of intersection, if any, of the graphs of the functions. Round your result to three decimal places. $$\begin{aligned} &y=20 e^{0.05 x}\\\ &y=1500 \end{aligned}$$

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{4} x y^{6} z^{4}$$.

Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{2} x, \quad g(x)=-3+\log _{2} x$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$4^{-2 t}=0.0625$$

Use a graphing utility to find the point(s) of intersection, if any, of the graphs of the functions. Round your result to three decimal places. $$\begin{aligned} &y=100 e^{0.01 x}\\\ &y=12,500 \end{aligned}$$

At 8: 30 A.M., a coroner was called to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person's temperature twice. At 9: 00 A.M. the temperature was \(85.7^{\circ} \mathrm{F},\) and at 11: 00 A.M. the temperature was \(82.8^{\circ} \mathrm{F}\). From these two temperatures the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula $$t=-10 \ln \frac{T-70}{98.6-70}$$ where \(t\) is the time (in hours elapsed since the person died) and \(T\) is the temperature (in degrees Fahrenheit) of the person's body. Assume that the person had a normal body temperature of \(98.6^{\circ} \mathrm{F}\) at death and that the room temperature was a constant \(70^{\circ} \mathrm{F}\). Use the formula to estimate the time of death of the person. (This formula is derived from a general cooling principle called Newton's Law of Cooling.)

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\ln \sqrt[3]{\frac{x^{4}}{y^{3}}}$$.

Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{8} x, \quad g(x)=-2+\log _{8}(x-3)$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$250 e^{0.02 x}=10,000$$

Use a graphing utility to find the point(s) of intersection, if any, of the graphs of the functions. Round your result to three decimal places. $$\begin{aligned} &y=2.5 e^{0.3 x}\\\ &y=0.2 \end{aligned}$$

You take a five-pound package of steaks out of a freezer at 11 A.M. and place it in a refrigerator. Will the steaks be thawed in time to be grilled at 6 P.M.? Assume that the refrigerator temperature is \(40^{\circ} \mathrm{F}\) and the freezer temperature is \(0^{\circ} \mathrm{F}\). Use the formula (derived from Newton's Law of Cooling) $$t=-5.05 \ln \frac{T-40}{0-40}$$ where \(t\) is the time in hours (with \(t=0\) corresponding to 11 A.M.) and \(T\) is the temperature of the package of steaks (in degrees Fahrenheit).

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\ln \sqrt{\frac{x^{2}}{y^{3}}}$$.

Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{8} x, \quad g(x)=4+\log _{8}(x-1)$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$100 e^{0.005 x}=125,000$$

Use a graphing utility to find the point(s) of intersection, if any, of the graphs of the functions. Round your result to three decimal places. $$\begin{aligned} &y=3.2 e^{0.5 x}\\\ &y=0.9 \end{aligned}$$

Determine whether the statement is true or false. Justify your answer. The domain of a logistic growth function cannot be the set of real numbers.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\ln \frac{x^{2}-1}{x^{3}}, \quad x>1$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln 6=1.7917 . . .$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$500 e^{-x}=300$$

(a) use a graphing utility to graph the function, (b) use the graph to find the open intervals on which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values. $$f(x)=x^{2} e^{-x}$$

Determine whether the statement is true or false. Justify your answer. The graph of a logistic growth function will always have an \(x\) -intercept.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\ln \frac{x}{\sqrt{x^{2}+1}}$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln 4=1.3862 . . .$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$1000 e^{-4 x}=75$$

(a) use a graphing utility to graph the function, (b) use the graph to find the open intervals on which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values. $$f(x)=2 x^{2} e^{x+1}$$

Can the graph of a Gaussian model ever have an \(x\) -intercept? Explain.

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{b} \frac{x^{4} \sqrt{y}}{z^{5}}$$.