91Ó°ÊÓ

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \sqrt{x^{2}+1} $$

Find the effective rate of interest corresponding to a nominal rate of \(9 \%\) per year compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly.

In Exercises, use a graphing utility to graph the function. Determine any asymptotes of the graph. $$ f(x)=\frac{8}{1+e^{-0.5 x}} $$

In Exercises, use a graphing utility to graph the function. Determine any asymptotes of the graph. $$ g(x)=\frac{8}{1+e^{-0.5 / x}} $$

In Exercises, use the properties of logarithms to write the expression as a sum, difference, or multiple of logarithms. $$ \ln \sqrt{\frac{x^{3}}{x+1}} $$

The effective yield is the annual rate \(i\) that will produce the same interest per year as the nominal rate \(r\). (a) For a rate \(r\) that is compounded continuously, show that the effective yield is \(i=e^{r}-1\). (b) Find the effective yield for a nominal rate of \(6 \%\), compounded continuously.

Find the effective rate of interest corresponding to a nominal rate of \(7.5 \%\) per year compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly.

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{7} \frac{2}{9} $$

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{1 / 5} 31 $$

In Exercises, solve the equation for \(x\). $$ e^{-3 x}=e $$