College Algebra, Chapter 5, 5.4, Section 5.4, Problem 30

Solve the equation $e^{2x} - e^x - 6 = 0$

$
\begin{equation}
\begin{aligned}
e^{2x} - e^x - 6 &= 0\\
\\
(e^x - 3)(e^x + 2) &= 0 && \text{Using trial and error}
\end{aligned}
\end{equation}
$

Solve for $x$

$
\begin{equation}
\begin{aligned}
e^x - 3 &= 0 &&& e^x + 2 &= 0\\
\\
e^x &= 3 &&\text{Add }3& e^x &= -2 && \text{Subtract 2}\\
\\
\ln e^x &= \ln 3 &&\text{Take $\ln$ of each side}& \ln e^x &= \ln(-2) && \text{Take $\ln$ of each side}\\
\\
x &= \ln3 &&\text{Property of $\ln$}& x &= \ln(-2) && \text{Property of $\ln$}
\end{aligned}
\end{equation}
$

The solution to the given equation is only $x = \ln 3$, since $\ln (-2)$ is undefined.