  Given an ideal $\mathcal I$ on a variety $X$ with toroidal singularities, we
produce a modification $X' \to X$, functorial for toroidal morphisms, making
the ideal monomial on a toroidal stack $X'$. We do this by adapting the methods
of [W{\l}o05], discarding steps which become redundant.
  We deduce functorial resolution of singularities for varieties with
logarithmic structures. This is the first step in our program to apply
logarithmic desingularization to a morphism $Z \to B$, aiming to prove
functorial semistable reduction theorems.
