Logarithmic generating function

Question

I'm fairly new to generating functions, but I'm encountering some sums that I think are amenable to logarithmic generating functions. One particular sum has the form $$f(x)=\sum_{n}\frac{f_{2n}}{n}x^{2n}$$ In this case I know what $g(x)=\sum_{n}f_{2n}x^{2n}$ sums to, and I can derive an expression for $f(x)$ by substituting $x\rightarrow\sqrt{x}$ into $g(x)$ and integrating on $x$, but I also know that $f(x)$ should be a logarithm of a finite polynomial in $g$. Is it possible to use some composition theorem to show that actually $f(x)=\ln P(g(x))$ without the substitutions and integrations?