Solve y=xlnx-a^x

$$y = x \ln x - a ^ { x }$$

Solve for a (complex solution)

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

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

$$x\neq 0$$

Solve for a

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