Answer
$$ x=\frac{-1}{2}\ln (e-1)$$
Work Step by Step
Since
\begin{align*}
e-e^{-2 x}&=1 \\
e-1&=e^{-2 x} ,\ \ \ \ \ \ \text{Take } \ \ln \ \text{for both sides }\\
\ln (e-1)&=\ln e^{-2 x} \\
-2 x=\ln (e-1)
\end{align*}
Then
$$ x=\frac{-1}{2}\ln (e-1)$$
