Question

$$x g ^ { x } = 729$$

Solve for g (complex solution)

$g=e^{-\frac{2\pi n_{1}iRe(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}-\frac{2\pi n_{1}Im(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}+\frac{arg(\frac{1}{x})Im(x)+iarg(\frac{1}{x})Re(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}\times 3^{\frac{6\left(Re(x)-iIm(x)\right)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}\left(|x|\right)^{\frac{-Re(x)+iIm(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}$
$n_{1}\in \mathrm{Z}$
$x\neq 0$

Solve for g

$\left\{\begin{matrix}g=\left(\frac{729}{x}\right)^{\frac{1}{x}}\text{, }&\left(Numerator(x)\text{bmod}2=1\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x<0\right)\text{ or }\left(\left(\frac{729}{x}\right)^{\frac{1}{x}}>0\text{ and }x>0\right)\text{ or }\left(\left(\frac{729}{x}\right)^{\frac{1}{x}}<0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x>0\right)\\g=-\left(\frac{729}{x}\right)^{\frac{1}{x}}\text{, }&\left(Numerator(x)\text{bmod}2=1\text{ and }x<0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(\left(\frac{729}{x}\right)^{\frac{1}{x}}>0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\text{ and }x>0\right)\text{ or }\left(\left(\frac{729}{x}\right)^{\frac{1}{x}}<0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }x>0\right)\end{matrix}\right.$