Solve y=3x^n+1

$$y = 3 x ^ { n + 1 }$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\frac{\ln(y)-\ln(x)-\ln(3)}{\ln(x)}+\frac{2\pi n_{1}i}{\ln(x)}\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 0\text{ and }x\neq 1\text{ and }x\neq 0\\n\in \mathrm{C}\text{, }&\left(x=0\text{ and }y=0\right)\text{ or }\left(x=1\text{ and }y=3\right)\end{matrix}\right.$$

Solve for x (complex solution)

$$x=\left(\frac{|y|}{3}\right)^{\frac{Re(n)\left(Im(n)\right)^{2}-iIm(n)\left(Re(n)\right)^{2}+\left(Re(n)\right)^{3}-i\left(Im(n)\right)^{3}-2iRe(n)Im(n)+3\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}+3Re(n)-iIm(n)+1}{\left(\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}+2Re(n)+1\right)^{2}}}e^{\frac{Im(n)arg(y)+iRe(n)arg(y)+iarg(y)}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}+2Re(n)+1}-\frac{2\pi n_{1}iRe(n)}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}+2Re(n)+1}-\frac{2\pi n_{1}Im(n)}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}+2Re(n)+1}-\frac{2\pi n_{1}i}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}+2Re(n)+1}}$$

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

Solve for n

$$\left\{\begin{matrix}n=-\frac{\ln(x)-\ln(y)+\ln(3)}{\ln(x)}\text{, }&y>0\text{ and }x\neq 1\text{ and }x>0\\n\in \mathrm{R}\text{, }&\left(x=-1\text{ and }y=-3\text{ and }Numerator(n+1)\text{bmod}2=1\text{ and }Denominator(n)\text{bmod}2=1\right)\text{ or }\left(y=3\text{ and }x=1\right)\\n>-1\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\left(\frac{y}{3}\right)^{\frac{1}{n+1}}\text{, }&\left(Numerator(n+1)\text{bmod}2=1\text{ and }n\neq -1\text{ and }Denominator(n)\text{bmod}2=1\text{ and }y<0\text{ and }\left(\frac{y}{3}\right)^{\frac{1}{n+1}}\neq 0\right)\text{ or }\left(\left(\frac{y}{3}\right)^{\frac{1}{n+1}}<0\text{ and }y>0\text{ and }n\neq -1\text{ and }Denominator(n)\text{bmod}2=1\right)\text{ or }\left(y=0\text{ and }n>-1\right)\text{ or }\left(\left(\frac{y}{3}\right)^{\frac{1}{n+1}}>0\text{ and }y>0\text{ and }n\neq -1\right)\\x=-\left(\frac{y}{3}\right)^{\frac{1}{n+1}}\text{, }&\left(y<0\text{ and }Numerator(n+1)\text{bmod}2=1\text{ and }n\neq -1\text{ and }Numerator(n+1)\text{bmod}2=0\text{ and }Denominator(n)\text{bmod}2=1\text{ and }\left(\frac{y}{3}\right)^{\frac{1}{n+1}}\neq 0\right)\text{ or }\left(y>0\text{ and }n\neq -1\text{ and }\left(\frac{y}{3}\right)^{\frac{1}{n+1}}>0\text{ and }Numerator(n+1)\text{bmod}2=0\text{ and }Denominator(n)\text{bmod}2=1\right)\text{ or }\left(Numerator(n+1)\text{bmod}2=0\text{ and }y=0\text{ and }n>-1\right)\text{ or }\left(y>0\text{ and }n\neq -1\text{ and }\left(\frac{y}{3}\right)^{\frac{1}{n+1}}<0\text{ and }Numerator(n+1)\text{bmod}2=0\right)\\x\neq 0\text{, }&n=-1\text{ and }y=3\end{matrix}\right.$$