91Ó°ÊÓ

Show that${\log }_a\left(x+\sqrt{x^2-1}\right)+{\log }_a\left(x-\sqrt{x^2-1}\right)=0$.

Solve the equation.

$e^{2x+5}=8$

$p\left(n\right)=100{(0.97)}^n$

(a) What percent of light will pass through 10 panes?
(b) What percent of light will pass through 25 panes?

Fill in reasons for each step in the following two solutions.

Solve: ${\log }_3{\left(x-1\right)}^2=2$

Solution A

$$\begin{gathered} {\log }_3{\left(x-1\right)}^2=2 \\ {\left(x-1\right)}^2=3^2=9\_\_\_\_\_\_\_\_\_\_\_ \\ \left(x-1\right)=Â\pm 3\_\_\_\_\_\_\_\_\_\_\_ \\ x-1=-3orx-1=3\_\_\_\_\_\_\_\_\_\_ \\ x=-2orx=4\_\_\_\_\_\_\_\_\_\_ \end{gathered}$$

Solution B

$$\begin{gathered} {\log }_3{\left(x-1\right)}^2=2 \\ 2{\log }_3\left(x-1\right)=2\_\_\_\_\_\_\_\_\_\_ \\ {\log }_3\left(x-1\right)=1\_\_\_\_\_\_\_\_\_\_\_ \\ x-1=3^1=3\_\_\_\_\_\_\_\_\_\_ \\ x=4\_\_\_\_\_\_\_\_\_\_ \end{gathered}$$

Both solutions given in Solution A check. Explain what caused the solution$x=-2$to be lost in Solution B.

Is every odd function one-to-one? Explain.

Solve the equation.

$e^{-2x+1}=13$

Show that${\log }_a\left(\sqrt{x}+\sqrt{x-1}\right)+{\log }_a\left(\sqrt{x}-\sqrt{x-1}\right)=0$.

Suppose that C(g) represents the cost C, in dollars, of manufacturing gcars. Explain what $C^{-1}(800,000)$represents.

$p\left(h\right)=760e^{-0.145h}$

(a) Find the atmospheric pressure at a height of 2 kilometers (over a mile).
(b) What is it at a height of 10 kilometers (over 30,000 feet)?

Show that$\ln \left(1+e^{2x}\right)=2x+\ln \left(1+e^{-2x}\right)$.