91Ó°ÊÓ

Determine the value of the unknown. $$\log _{3} 27^{-1}=x+1$$

Use logarithms to perform the indicated calculations. A certain type of optical switch in a fiber-optic system allows a light signal to continue in either of two fibers. How many possible paths could a light signal follow if it passes through 400 such switches?

The strength \(I\) of a certain cable signal is given by \(I=I_{0} e^{-0.0015 x}\) where \(I_{0}\) is the signal strength at the source and \(x\) is the distance (in \(\mathrm{km}\) ) from the source. What percent of the signal strength is lost 15 km from the source?

Express each as a sum, difference, or multiple of logarithms. In each case, part of the logarithm may be determined exactly. $$3 \log _{10}\left(40^{2}\right)$$

Express \(\ln \left(x e^{-x}\right)\) as the sum or difference of logarithms, evaluating where possible.

The flash unit on a camera operates by releasing the stored charge on a capacitor. For a particular unit, the charge \(q\) (in \(\mu \mathrm{C}\) ) as a function of the time \(t\) (in s) is \(q=100 e^{-10 t} .\) Display the graph on a calculator.

Plot the graphs of the given functions. $$y=\log _{3} x$$

Use a calculator to solve the given equations. Solve for \(x: e^{x}+e^{-x}=3 .\) (Hint: Multiply each term by \(e^{x}\) and then it can be treated as a quadratic equation in \(e^{x}\).)

Solve for \(y\) in terms of \(x\) $$\log _{b} y=\log _{b} 2+\log _{b} x$$

Express \(\ln \left(e^{2} \sqrt{1-x}\right)\) as the sum or difference of logarithms, evaluating where possible.