I read in a paper of Kato about log-differential forms, that if $X$ is a smooth locally Noetherian log-scheme, and $D$ is a reduced normal crossing divisor, then there is a definition of a sheaf on $X$, denoted $\Omega^1_X(\log D)$ of $\log$-differential forms, which has a seemingly convoluted definition in terms of a quotient of $\Omega^1_X\bigoplus (M^{gp}\otimes \mathcal{O}_X)$, where $M$ is the monoidal sheaf on $X$ corresponding to the standard log-structure on $X$ with respect to $D$, and $M^{gp}$ is the group completion of the monoidal sheaf $M$.
When $D$ is a hypersurface cut out locally by the zero set of a function $f$ on $X$, I know there is a local description of the sections of $\Omega_X^1(\log D)$ via $\omega/f$, where $\omega$ is a local section of the sheaf $\Omega_X^1$, i.e. of the usual sheaf of 1-forms on $X$ in the algebraic geometric sense.
Is there a way to see that the two definitions are equivalent in this case? A reference would also be appreciated.
