Calculus: Early Transcendentals, Chapter 1, Review Exercises, Section Review Exercises, Problem 26

Evaluate each equation for $x$
a.) $e^x = 5$
b.) $\ln x = 2$
c.) $e^{e^x} = 2$
d.) $\tan^{-1} x = 1$

a.) $e^x = 5$

$
\begin{equation}
\begin{aligned}
\ln e^x &= \ln 5
&& \text{Take ln of each side}\\
\\
x \cdot \ln e &= \ln 5
&& \text{Use the property of ln}\\
\\
x \cdot 1 &= \ln 5
&& \text{Simplify}\\
\\
x &= \ln 5
\end{aligned}
\end{equation}
$


b.) $\ln x = 2$

$
\begin{equation}
\begin{aligned}
e^{\ln x} &= e^2
&& \text{Raise both sides in } e\\
\\
x &= e^2
&& \text{Simplify}
\end{aligned}
\end{equation}
$


c.) $e^{e^x} = 2$

$
\begin{equation}
\begin{aligned}
\ln e^{e^x} &= \ln 2
&& \text{Take ln of each side}\\
\\
e^x &= \ln 2
&& \text{Use the property of ln}\\
\\
\ln e^x &= \ln (\ln 2)
&& \text{Again, take ln of each side}\\
\\
x &= \ln (\ln 2)
&& \text{Solve for } x
\end{aligned}
\end{equation}
$


d.) $\tan^{-1} x = 1$
$x = \tan (1)$