Solve y=c^2x

$$y = c ^ { 2 x }$$

Solve for c (complex solution)

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

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

Solve for x (complex solution)

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

Solve for c

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