Chapter 4

Use the properties of logarithms to condense the expression.$$\frac{1}{2} \ln \left(x^{2}+4\right)+\ln x$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$\sqrt{e^{3}}=4.4816. . . $$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$e^{x^{2}-3 x}=e^{x-2}$$

You build an annuity by investing \(P\) dollars every month at interest rate \(r,\) compounded monthly. Find the amount \(A\) accrued after \(n\) months using the formula. \(A=P\left[\frac{(1+r / 12)^{n}-1}{r / 12}\right],\) where \(r\) is in decimal form. $$P=S 200, r=0.06, n=72 \text { months }$$

Use the Leading Coefficient Test to determine the right-hand and left-hand behavior of the graph of the polynomial function. $$g(x)=7 x^{6}+9.1 x^{5}-3.2 x^{4}+25 x^{3}$$

Use the properties of logarithms to condense the expression.$$2 \ln x+\ln (x+1)$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$e^{3 / 4}=2.1170. . . $$

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

You build an annuity by investing \(P\) dollars every month at interest rate \(r,\) compounded monthly. Find the amount \(A\) accrued after \(n\) months using the formula. \(A=P\left[\frac{(1+r / 12)^{n}-1}{r / 12}\right],\) where \(r\) is in decimal form. $$P=\$ 75, r=0.03, n=24 \text { months }$$

Divide using synthetic division. $$\left(2 x^{3}-8 x^{2}+3 x-9\right) \div(x-4)$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=11$$ $$x=18.31$$ $$x=\frac{1}{2}$$ $$x=\sqrt{0.65}$$ (Function) $$f(x)=\ln x$$

Use the properties of logarithms to condense the expression.$$\ln x-3 \ln (x+1)$$.

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$\frac{400}{1+e^{-x}}=350$$

There are three options for investing \(\$ 500 .\) The first earns \(7 \%\) compounded annually, the second earns \(7 \%\) compounded quarterly, and the third earns \(7 \%\) compounded continuously. (a) Find equations that model the growth of each investment and use a graphing utility to graph each model in the same viewing window over a 20-year period. (b) Use the graph from part (a) to determine which investment yields the highest return after 20 years. What are the differences in earnings among the three investments?

Divide using synthetic division. $$\left(x^{4}-3 x+1\right) \div(x+5)$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=11$$ $$x=18.31$$ $$x=\frac{1}{2}$$ $$x=\sqrt{0.65}$$ (Function) $$f(x)=\ln x$$

Use the properties of logarithms to condense the expression.$$\ln x-2 \ln (x+2)$$.

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$\frac{525}{1+e^{-x}}=275$$

Let \(Q\) represent a mass, in grams, of radioactive radium ( \(226 \mathrm{Ra}\) ), whose half-life is 1600 years. The quantity of radium present after \(t\) years is given by \(Q=25\left(\frac{1}{2}\right)^{\gamma / 1600}\) (a) Determine the initial quantity (when \(t=0\) ). (b) Determine the quantity present after 1000 years. (c) Use a graphing utility to graph the function over the interval \(t=0\) to \(t=5000\) (d) When will the quantity of radium be 0 grams? Explain.

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. (Value) $$x=11$$ $$x=18.31$$ $$x=\frac{1}{2}$$ $$x=\sqrt{0.65}$$ (Function) $$f(x)=-\ln x$$

Use the properties of logarithms to condense the expression.$$\ln (x-2)+\ln 2-3 \ln y$$.

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$\frac{40}{1-5 e^{-0.01 x}}=200$$

Let \(Q\) represent a mass, in grams, of carbon \(14\left(^{14} \mathrm{C}\right),\) whose half-life is 5700 years. The quantity present after \(t\) years is given by \(Q=10\left(\frac{1}{2}\right)^{i / 5700}\) (a) Determine the initial quantity (when \(t=0\) ). (b) Determine the quantity present after 2000 years. (c) Sketch the graph of the function over the interval \(t=0\) to \(t=10,000\)

Use the properties of logarithms to condense the expression.$$3 \ln x+2 \ln y-4 \ln z$$.

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$\frac{50}{1-2 e^{-0.001 x}}=1000$$

Assume the annual rate of inflation is \(4 \%\) for the next 10 years. The approximate cost \(C\) of goods or services during these years is \(C(t)=P(1.04)^{t},\) where \(t\) is the time (in years) and \(P\) is the present cost. An oil change for your car presently costs \(\$ 26.88 .\) Use the following methods to approximate the cost 10 years from now. (a) Use a graphing utility to graph the function and then use the value feature. (b) Use the table feature of the graphing utility to find a numerical approximation. (c) Use a calculator to evaluate the cost function algebraically.

Use the properties of natural logarithms to rewrite the expression. $$\ln e^{2}$$

Use the properties of logarithms to condense the expression.$$\ln x-2[\ln (x+2)+\ln (x-2)]$$.

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$e^{3 x}=12$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 0.6 & 0.7 & 0.8 & 0.9 & 1.0 \\\\\hline e^{3 x} & & & & & \\\\\hline\end{array}$$

The projected populations of California for the years 2020 through 2060 can be modeled by \(P=36.308 e^{0.0065 t},\) where \(P\) is the population (in millions) and \(t\) is the time (in years), with \(t=20\) corresponding to \(2020 .\) (Source: California Department of Finance) (a) Use a graphing utility to graph the function for the years 2020 through 2060 (b) Use the table feature of the graphing utility to create a table of values for the same time period as in part (a). (c) According to the model, in what year will the population of California exceed 51 million?

Use the properties of natural logarithms to rewrite the expression. $$-\ln e$$

Use the properties of logarithms to condense the expression.$$4[\ln z+\ln (z+5)]-2 \ln (z-5)$$.

\((p .324)\) In early \(2014,\) a new sedan had a manufacturer's suggested retail price of \(\$ 31,340 .\) After \(t\) years, the sedan's value is given by \(V(t)=31,340\left(\frac{4}{5}\right)^{t}\) (a) Use a graphing utility to graph the function. (b) Use the graphing utility to create a table of values that shows the value \(V\) for \(t=1\) to \(t=10\) years. (c) According to the model, when will the sedan have no value?

Use the properties of natural logarithms to rewrite the expression. $$e^{\ln 1.8}$$

Use the properties of logarithms to condense the expression.$$\frac{1}{3}\left[2 \ln (x+3)+\ln x-\ln \left(x^{2}-1\right)\right]$$.

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$20\left(100-e^{x / 2}\right)=500$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 5 & 6 & 7 & 8 & 9 \\\\\hline 20\left(100-e^{x / 2}\right) & & & & & \\\\\hline\end{array}$$

Determine whether the statement is true or false. Justify your answer. \(f(x)=1^{x}\) is not an exponential function.

Use the properties of natural logarithms to rewrite the expression. $$7 \ln e^{0}$$

Use the properties of logarithms to condense the expression.$$2[\ln x-\ln (x+1)-\ln (x-1)]$$.

(a) complete the table to find an interval containing the solution of the equation, (b) use a graphing utility to graph both sides of the equation to estimate the solution, and (c) solve the equation algebraically. Round your results to three decimal places. $$11\left(77-e^{x-4}\right)=264$$ $$\begin{array}{|l|l|l|l|l|l|}\hline x & 5 & 6 & 7 & 8 & 9 \\\\\hline 11\left(77-e^{x-4}\right) & & & & & \\\\\hline\end{array}$$

Use the properties of natural logarithms to rewrite the expression. $$e \ln 1$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$y_{1}=2\left[\ln 8-\ln \left(x^{2}+1\right)\right], \quad y_{2}=\ln \left[\frac{64}{\left(x^{2}+1\right)^{2}}\right]$$

Use the zero or root feature or the zoom and trace features of a graphing utility to approximate the solution of the exponential equation accurate to three decimal places. $$\left(1+\frac{0.065}{365}\right)^{365 t}=4$$

Use the properties of natural logarithms to rewrite the expression. $$e^{\ln 22}$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$y_{1}=2\left[\ln 6+\ln \left(x^{2}+1\right)\right], \quad y_{2}=\ln \left[36\left(x^{2}+1\right)^{2}\right]$$.

Use the zero or root feature or the zoom and trace features of a graphing utility to approximate the solution of the exponential equation accurate to three decimal places. $$\left(4-\frac{2.471}{40}\right)^{9 t}=21$$

Exploration Use a graphing utility to graph \(y_{1}=e^{x}\) and each of the functions \(y_{2}=x^{2}, y_{3}=x^{3}, y_{4}=\sqrt{x}\) and \(y_{5}=|x|\) in the same viewing window. (a) Which function increases at the fastest rate for "large" values of \(x ?\) (b) Use the result of part (a) to make a conjecture about the rates of growth of \(y_{1}=e^{x}\) and \(y=x^{n},\) where \(n\) is a natural number and \(x\) is "large." (c) Use the results of parts (a) and (b) to describe what is implied when it is stated that a quantity is growing exponentially.

Use the properties of natural logarithms to rewrite the expression. $$\ln e^{\ln e}$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$y_{1}=\ln x+\frac{1}{2} \ln (x+1), \quad y_{2}=\ln (x \sqrt{x+1})$$

Use the zero or root feature or the zoom and trace features of a graphing utility to approximate the solution of the exponential equation accurate to three decimal places. $$\frac{7000}{5+e^{3 x}}=2$$