Solve a^x=n

$${ a }^{ x } = n$$

Solve for a (complex solution)

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

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

Solve for n (complex solution)

$$n=a^{x}$$

Solve for a

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

Solve for n

$$n=a^{x}$$

$$\left(a<0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(a=0\text{ and }x>0\right)\text{ or }a>0$$