Solve varepsilon=I^R

$$\varepsilon = I ^ { R }$$

Solve for I (complex solution)

$$I=e^{\frac{Im(R)arg(\epsilon )+iRe(R)arg(\epsilon )}{\left(Re(R)\right)^{2}+\left(Im(R)\right)^{2}}-\frac{2\pi n_{1}iRe(R)}{\left(Re(R)\right)^{2}+\left(Im(R)\right)^{2}}-\frac{2\pi n_{1}Im(R)}{\left(Re(R)\right)^{2}+\left(Im(R)\right)^{2}}}\left(|\epsilon |\right)^{\frac{Re(R)-iIm(R)}{\left(Re(R)\right)^{2}+\left(Im(R)\right)^{2}}}$$

$$n_{1}\in \mathrm{Z}$$

Solve for R (complex solution)

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

Solve for I

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

Solve for R

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