Is it possible to generalise the values of $a$ and $b$ for these following expressions?

Question

$a{}^b$ = m

$b{}^a$ = n

Find $a$ and $b$ in terms of $m$ and $n$

Such that $a,b,m,n$ $\in {}^+R$

I tried logging both sides and substituting the values. It forms an equation $n{}^\frac{1}{a} \ln (a) - \ln (m) = 0$