Solve y=a^x-2+1

$$y = a ^ { x - 2 } + 1$$

Solve for a (complex solution)

$$a=e^{\frac{Im(x)arg(-\left(1-y\right))+iRe(x)arg(-\left(1-y\right))-2iarg(-\left(1-y\right))}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}-4Re(x)+4}-\frac{2iRe(x)\pi n_{1}}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}-4Re(x)+4}-\frac{2\pi n_{1}Im(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}-4Re(x)+4}+\frac{4i\pi n_{1}}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}-4Re(x)+4}}\left(|1-y|\right)^{\frac{Re(x)-iIm(x)-2}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}-4Re(x)+4}}$$

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

Solve for x (complex solution)

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

Solve for a

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

Solve for x

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