Answer
$x=2e^4+1$
Work Step by Step
Recall:
$$\ln{a}=y \longleftrightarrow e^y=a$$
Use the definition above to obtain:
\begin{align*}
\ln{\frac{x-1}{2}}&=4\\\\
e^{4}&=\frac{x-1}{2}\\\\
2\left(e^{4}\right)&=2\left(\frac{x-1}{2}\right)\\\\
2e^4&=x-1\\\\
2e^4+1&=x\end{align*}
Check:
\begin{align*}
\ln{\frac{2e^4+1-1}{2}}&\stackrel{?}=4\\\\
\ln{\frac{2e^4}{2}}&\stackrel{?}=4\\\\
\ln{e^4}&\stackrel{?}=4\\\\
4&\stackrel{\checkmark}=4
\end{align*}
