Is there a closed form? Is there a closed form for $k$ in the expression
$$am^k + bn^k = c$$
where $a, b, c, m, n$ are fixed real numbers?
If there is no closed form, what other ways are there of finding $k$?
Motivation: It came up when trying to apply an entropy model to allele distribution in genetics. The initial population sizes are $a$ and $b$, and get decayed by $m, n < 1$ respectively $k$ times until the population drops to the carrying capacity $c$.

I believe a closed form would only be possible if you could express $n^k$ in terms of $m^k$. Otherwise you would need to find the roots of $am^k + bn^k -c$, which I don't think is possible with elementary functions.
An well-known way to approximate roots is with Newton's Method.
