If $x^ay^b=e^m, x^cy^d = e^n, Δ_1=\begin{vmatrix}m&b\\n&d\end{vmatrix},Δ_2=\begin{vmatrix}a&m\\c&n\end{vmatrix}$ and $Δ_3=\begin{vmatrix}a&b\\c&d\end{vmatrix}$, then the values of x and y are

$e^{Δ_1/Δ_3}$ and $e^{Δ_2/Δ_3}$

We have,

$x^ay^b = e^m,x^cy^d = e^n$

$⇒a \log x + b \log y=m$

$c \log x + d \log y=n$

Using Cramer's rule, we have

$\log x=\frac{\begin{vmatrix}m&b\\n&d\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}$ and $\log y = \frac{\begin{vmatrix}a&m\\c&n\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}$

$⇒\log x=\frac{Δ_1}{Δ_3}$ and $\log y =\frac{Δ_2}{Δ_3}$

$⇒x=e^{Δ_1/Δ_3}$ and $y =e^{Δ_2/Δ_3}$