I ask about a possible method to find the solution of algebraic equations of the form


$axⁿ+byⁿ+c=0$


where $a,b,c,x,y$ are real constants and $n$ is an integer. Maybe there is a simple method, but I cannot find it.



Solution of certain forms of equations
If $\log\_x(y) = j/k$ is rational, this reduces to a polynomial in $x^{1/k} = y^{1/j}$.
Otherwise you're unlikely to get a closed form. You might use numerical methods, or
a series expansion: if $y = x^r$,
$$ n = \frac{\ln(-c/a)}{\ln(x)} + \sum\_{k=0}^\infty \frac{(-c/a)^{kr}(b/c)^k}{k! \ln(x)} \prod\_{j=1}^{k-1} (kr - j)$$


