Resolver y=x^ne

$$y = x ^ { n } e$$

Resolva para n (complex solution)

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

Resolva para x (complex solution)

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

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

Resolva para n

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

Resolva para x

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