The Lambert W function, or product logarithm, is closely related to the logarithm. Specifically, the logarithm is the inverse of $f(x) = \mathrm{e}^x$, whereas the Lambert W function is the inverse of $g(x) = x\mathrm{e}^x$ (see https://en.wikipedia.org/wiki/Lambert_W_function).
Is there any unit of measure for the Lambert W function in information theory?
The reason I am asking is that, in information theory, the information content of a message having probability $p$ is $-\log_2 (p)$ bits if all messages are equally likely. Is there a unit of measure (information-theoretic or otherwise) for quantities expressed in terms of the Lambert W function?