Inverse Baker-Campbell-Hausdorff formula

Statement
The inverse Baker-Campbell-Hausdorff formula is a formula that tries to express:

$$\! \exp(\log(X) + \log(Y))$$

in terms of $$X$$ and $$Y$$. Here $$X$$ and $$Y$$ are elements of a Lie group (or analogue thereof), $$\log$$ is the logarithm map to its Lie ring or Lie algebra, addition happens in the Lie ring, and $$\exp$$ is the exponential map.

Note that there is a related inverse formula which is the formula for Lie bracket in terms of group commutator.