  We use filtered log-$\mathscr{D}$-modules to represent the (dual)
localization of Saito's Mixed Hodge Modules along a smooth hypersurface, and
show that they also behave well under the direct image functor and the dual
functor in the derived category of filtered log-$\mathscr{D}$-modules. The
results of this paper can be used to generalize the result of M. Popa and C.
Schnell about Kodaira dimension and zeros of holomorphic one-forms into the log
setting.
