Solve y=(x+2)^R

$$y = ( x + 2 ) ^ { R }$$

Solve for R (complex solution)

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

Solve for x (complex solution)

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

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

Solve for R

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

Solve for x

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