Solve n^x/7=y

$$\frac { n ^ { x } } { 7 } = y$$

Solve for n (complex solution)

$$n=e^{\frac{Im(x)arg(y)+iRe(x)arg(y)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}-\frac{2\pi n_{1}iRe(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}-\frac{2\pi n_{1}Im(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}\times \left(7|y|\right)^{\frac{Re(x)-iIm(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}$$

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\ln(y)+\ln(7)}{\ln(n)}+\frac{2\pi n_{1}i}{\ln(n)}\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 0\text{ and }n\neq 1\text{ and }n\neq 0\\x\in \mathrm{C}\text{, }&\left(n=0\text{ and }y=0\right)\text{ or }\left(n=1\text{ and }y=\frac{1}{7}\right)\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=\left(7y\right)^{\frac{1}{x}}\text{, }&\left(Numerator(x)\text{bmod}2=1\text{ and }Denominator(x)\text{bmod}2=1\text{ and }y<0\text{ and }\left(7y\right)^{\frac{1}{x}}\neq 0\right)\text{ or }\left(\left(7y\right)^{\frac{1}{x}}<0\text{ and }y>0\text{ and }x\neq 0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(x>0\text{ and }y=0\right)\text{ or }\left(\left(7y\right)^{\frac{1}{x}}>0\text{ and }y>0\text{ and }x\neq 0\right)\\n=-\left(7y\right)^{\frac{1}{x}}\text{, }&\left(y<0\text{ and }Numerator(x)\text{bmod}2=1\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }\left(7y\right)^{\frac{1}{x}}\neq 0\right)\text{ or }\left(y>0\text{ and }x\neq 0\text{ and }\left(7y\right)^{\frac{1}{x}}>0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(Numerator(x)\text{bmod}2=0\text{ and }y=0\text{ and }x>0\right)\text{ or }\left(y>0\text{ and }x\neq 0\text{ and }\left(7y\right)^{\frac{1}{x}}<0\text{ and }Numerator(x)\text{bmod}2=0\right)\\n\neq 0\text{, }&x=0\text{ and }y=\frac{1}{7}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\ln(y)+\ln(7)}{\ln(n)}\text{, }&y>0\text{ and }n\neq 1\text{ and }n>0\\x\in \mathrm{R}\text{, }&\left(n=1\text{ and }y=\frac{1}{7}\right)\text{ or }\left(n=-1\text{ and }y=-\frac{1}{7}\text{ and }Denominator(x)\text{bmod}2=1\text{ and }Numerator(x)\text{bmod}2=1\right)\\x>0\text{, }&n=0\text{ and }y=0\end{matrix}\right.$$