Solve x^a=2x^2a

$$x ^ { a } = 2 x ^ { 2 a }$$

Solve for a (complex solution)

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

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=e^{-\frac{2\pi n_{1}\left(Im(a)+iRe(a)\right)}{\left(Re(a)\right)^{2}+\left(Im(a)\right)^{2}}}\times 2^{\frac{-Re(a)+iIm(a)}{\left(Re(a)\right)^{2}+\left(Im(a)\right)^{2}}}\text{, }n_{1}\in \mathrm{Z}\text{, }&\text{unconditionally}\\x=0\text{, }&a\neq 0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a>0\text{, }&x=0\\a=-\log_{x}\left(2\right)\text{, }&x\neq 1\text{ and }x>0\end{matrix}\right.$$

Solve for x

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