Definitions and assumptions

On a 2n-dimensional phase-space with coordinates $x$ and $p$, the Moyal product can be written explicitly as

$$g(x,p) \star h(x,p) = g(x,p) e^{\frac{i}{2}\left( \overleftarrow{\partial_x} \cdot \overrightarrow{\partial_p} - \overrightarrow{\partial_x} \cdot \overleftarrow{\partial_p}\right)} h(x,p) \, .$$

We define the $\star$-exponential $e_\star$ as $$e_\star^{f(x,p)}=\sum_{n=0}^{\infty}\frac{f^{\star n}}{n!}$$

or more precisely as the solution to the differential equation $$\frac{d}{dt} F(x,p;t)=H(x,p)\star F(x,p;t) \, \, \, \& \, \, \, F(x,p;0)=1,$$ i.e. $F(x,p;t)=e_\star^{-itH(x,p)}$.

Problem

Suppose we have two phase-space functions, $f(x,p)$ and $g(x,p)$. Is it always possible to find a unique $h(x,p)$ such that $$e_\star^{f(x,p)} \star e_\star^{g(x,p)} = e_\star^{h(x,p)} \, \, ?$$

I tried to fiddle with it and I searched far and wide trying to dig out an answer or something that might point me to the answer or (even better) a procedure for calculating $h(x,p)$, but without any luck. Any ideas? Have you seen something like this?