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