Consider for example the function:
$$x^2 e^y + \log(x)y^2 = 0$$
I suspect that neither x nor y can be isolated in this function (that it, it cannot be written as either $x(y)$ or $y(x)$. However, I can't really prove that other than to say "it's difficult to do if not impossible".
Is there a method by which it can be proven that no finite number of algebraic operations will lead to such a simplification? (The above function was presented as an example, but I am looking for a general solution)