Solve x_{I}=-q^-x

$$x _ { I } = - q ^ { - x }$$

Solve for q (complex solution)

$$\left\{\begin{matrix}q=e^{-\frac{2\pi n_{1}iRe(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}-\frac{2\pi n_{1}Im(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}+\frac{arg(-\frac{1}{x_{I}})Im(x)+iarg(-\frac{1}{x_{I}})Re(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}\left(|x_{I}|\right)^{\frac{-Re(x)+iIm(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}\text{, }n_{1}\in \mathrm{Z}\text{, }&x_{I}\neq 0\\q=0\text{, }&x\neq 0\end{matrix}\right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{2\pi n_{1}i}{\ln(q)}-\log_{q}\left(-x_{I}\right)\text{, }n_{1}\in \mathrm{Z}\text{, }&x_{I}\neq 0\text{ and }q\neq 1\text{ and }q\neq 0\\x\in \mathrm{C}\text{, }&\left(q=0\text{ and }x_{I}=0\right)\text{ or }\left(q=1\text{ and }x_{I}=-1\right)\end{matrix}\right.$$

Solve for q

$$\left\{\begin{matrix}q=\left(-\frac{1}{x_{I}}\right)^{\frac{1}{x}}\text{, }&\left(Numerator(x)\text{bmod}2=1\text{ and }Denominator(x)\text{bmod}2=1\text{ and }\left(-\frac{1}{x_{I}}\right)^{\frac{1}{x}}\neq 0\text{ and }x_{I}>0\right)\text{ or }\left(\left(-\frac{1}{x_{I}}\right)^{\frac{1}{x}}>0\text{ and }x\neq 0\text{ and }x_{I}<0\right)\text{ or }\left(\left(-\frac{1}{x_{I}}\right)^{\frac{1}{x}}<0\text{ and }x\neq 0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x_{I}<0\right)\\q=-\left(-\frac{1}{x_{I}}\right)^{\frac{1}{x}}\text{, }&\left(x_{I}>0\text{ and }Numerator(x)\text{bmod}2=1\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }\left(-\frac{1}{x_{I}}\right)^{\frac{1}{x}}\neq 0\right)\text{ or }\left(x_{I}<0\text{ and }x\neq 0\text{ and }\left(-\frac{1}{x_{I}}\right)^{\frac{1}{x}}<0\text{ and }Numerator(x)\text{bmod}2=0\right)\text{ or }\left(x_{I}<0\text{ and }x\neq 0\text{ and }\left(-\frac{1}{x_{I}}\right)^{\frac{1}{x}}>0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\right)\\q\neq 0\text{, }&x_{I}=-1\text{ and }x=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\log_{q}\left(-\frac{1}{x_{I}}\right)\text{, }&x_{I}<0\text{ and }q\neq 1\text{ and }q>0\\x\in \mathrm{R}\text{, }&\left(q=-1\text{ and }x_{I}=1\text{ and }Denominator(x)\text{bmod}2=1\text{ and }Numerator(x)\text{bmod}2=1\right)\text{ or }\left(q=1\text{ and }x_{I}=-1\right)\end{matrix}\right.$$