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 $c$.

In general, if we wish to find a solution to an equation $f(x)=c$ and we can rewrite said equation as $x=g(x)$ where $g(x)$ is some function involving $c$. Then, if possible, we may find an interval $I$ on which $g(x)$ is a contraction. By the Banach fixed-point theorem, the sequence $x_0$, $x_n+1=g(x_n)$ where $x_0$ is an arbitrary point in $I$ converges to this fixed point. 
