Consider the power series ring in noncommutative associative formal variables $X, Y$ over the field $\mathbb C$. Let $P$ be a power series. Define the exponential $e^P$ and logarithm $\log (1+P)$ using the usual series expansions. The exponential maps from a neighborhood of $0$ to a neighborhood of $1$ and the logarithm goes the other way.
Let $P$ be such a power series with zero constant term. How to show that the logarithm and exponential maps defined above are inverses of each other?