Solve a^frac{y{2}}=a^frac{5y{2}}=

$$a ^ { \frac { y } { 2 } } = a ^ { \frac { 5 y } { 2 } } =$$

Solve for a (complex solution)

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

Solve for y (complex solution)

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

Solve for a

$$\left\{\begin{matrix}\\a=1\text{, }&\text{unconditionally}\\a=0\text{, }&y>0\\a=-1\text{, }&\left(Numerator(\frac{y}{2})\text{bmod}2=1\text{ and }Denominator(\frac{y}{2})\text{bmod}2=1\text{ and }Numerator(\frac{5y}{2})\text{bmod}2=1\text{ and }Denominator(\frac{5y}{2})\text{bmod}2=1\right)\text{ or }\left(Numerator(\frac{y}{2})\text{bmod}2=0\text{ and }Numerator(\frac{5y}{2})\text{bmod}2=0\text{ and }Denominator(\frac{y}{2})\text{bmod}2=1\text{ and }Denominator(\frac{5y}{2})\text{bmod}2=1\right)\\a\neq 0\text{, }&y=0\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y>0\text{, }&a=0\\y\in \mathrm{R}\text{, }&\left(a=-1\text{ and }Denominator(\frac{y}{2})\text{bmod}2=1\text{ and }Numerator(\frac{y}{2})\text{bmod}2=1\text{ and }Denominator(\frac{5y}{2})\text{bmod}2=1\right)\text{ or }a=1\\y=0\text{, }&a\neq 0\text{ and }|a|\neq 1\end{matrix}\right.$$