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 $c$.

A closed form solution can only exist if m is a rational power of n, and/or $abc=0$. If such is not the case, let $\gamma=\dfrac1{\ln m-\ln n},\quad\alpha=\dfrac cb,\quad\beta=-\dfrac ab$ . Then $k=-x$, where x is the solution to the recursive equation $x=\gamma\ln(\alpha m^x+\beta)$, which can be computed using the following iterative algorithm: $x_0=\ldots$ , and $x_{n+1}=\gamma\ln(\alpha m^{x_n}+\beta)$.
