Solve y=u^x

$$y = u ^ { x }$$

Solve for u (complex solution)

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

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

Solve for x (complex solution)

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

Solve for u

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

Solve for x

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