$n=-\frac{e^{x}-2}{e^{2x}}$

$\left\{\begin{matrix}x=-\left(-\ln(\frac{\sqrt{8n+1}-1}{n})+\ln(2)\right)\text{, }&n\neq 0\text{ and }n\geq -\frac{1}{8}\\x=\ln(-\frac{\sqrt{8n+1}+1}{2n})\text{, }&n\geq -\frac{1}{8}\text{ and }n<0\\x=\ln(2)\text{, }&n=0\end{matrix}\right.$