91Ó°ÊÓ

Assuming that the annual rate of inflation averages \(4 \%\) over the next 10 years, the approximate costs \(C\) of goods or services during any year in that decade will be modeled by \(C(t)=P(1.04)^{t},\) where \(t\) is the time in years and \(P\) is the present cost. The price of an oil change for your car is presently 23.95 dollar. Estimate the price 10 years from now.

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$g(x)=\ln (-x)$$

Condense the expression to the logarithm of a single quantity. $$\log _{5} 8-\log _{5} t$$

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$2 \ln (x+3)=3$$

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.

Find a logarithmic equation that relates \(y\) and \(x .\) Explain the steps used to find the equation. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline y & 0.5 & 2.828 & 7.794 & 16 & 27.951 & 44.091 \\ \hline\end{array}$$

You are investing \(P\) dollars at an annual interest rate of \(r,\) compounded continuously, for \(t\) years. Which of the following would result in the highest value of the investment? Explain your reasoning. (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years.

Let \(f(x)=\log _{a} x\) and \(g(x)=a^{x},\) where \(a>1\) (a) Let \(a=1.2\) and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of intersection of the two graphs. (b) Determine the value(s) of \(a\) for which the two graphs have one point of intersection. (c) Determine the value(s) of \(a\) for which the two graphs have two points of intersection.

Use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{2} x$$