Solve u=x^nln(y/x)

$$u = x ^ { n } \ln ( \frac { y } { x } )$$

Solve for n (complex solution)

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

Solve for u (complex solution)

$$u=\ln(\frac{y}{x})x^{n}$$

$$y\neq 0\text{ and }x\neq 0$$

Solve for n

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

Solve for u

$$u=\ln(\frac{y}{x})x^{n}$$

$$\left(Denominator(n)\text{bmod}2=1\text{ and }y<0\text{ and }x<0\right)\text{ or }\left(y>0\text{ and }x>0\right)$$