See [LSY15, page 354].
Definition
Let \(f:X\to Y\) be a morphism of schemes and assume that \(Y\) is endowed with a log structure \(\alpha_Y:\mathcal M_Y\to \mathcal O_Y\). The pullback log structure on \(X\), denoted by \(f^*\mathcal M_Y\), is the log structure associated to the pre-log structure defined by the composition
\begin{align*} f^{-1}(\mathcal M_Y)\xrightarrow{\alpha_Y}f^{-1}(\mathcal O_Y)\xrightarrow{f^*}\mathcal O_X. \end{align*}The pullback commutes with the ghost sheaf, in the sense that
\begin{align*} \overline{f^*\mathcal M_Y} = f^{-1}\overline{\mathcal M}_y. \end{align*}