The exponential formula (or polymer expansion, see https://en.wikipedia.org/wiki/Exponential_formula), allows one to transform a power series:
$$f(x) = a_1x+\frac{a_2}{2}x^2 + \dots = \sum_{n=1}^\infty \frac{a_n}{n!}x^n$$
into the exponential power series:
$$e^{f(x)} = \sum_{n=0}^\infty \frac{b_n}{n!}x^n$$
where
$$b_n = B_n(a_1,\dots,a_n)$$
are the Bell polynomials.
Is there a way to go the other way, from the exponential power series, to the original function?
More precisely, suppose that we know the coefficients $b_n$ in the expansion:
$$e^{f(x)} = \sum_{n=0}^\infty \frac{b_n}{n!}x^n$$
Can I compute the coefficients $a_n$ of $f(x)=\sum_{n=1}^\infty \frac{a_n}{n!}x^n$, in terms of the $b_n$? In other words, I would like to find the power series of the composition of $\log$ applied to $e^{f(x)}$. Is there a "logarithmic" version of the polymer expansion?