Is there a closed form?

Question

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 $c$.

Answer 1

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.

Answer 2

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 $x_{n+1}=\gamma\ln(\alpha m^{x_n}+\beta)$.

Answer 3

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.