Principalization of ideals on toroidal orbifolds

Given an ideal I on a variety X with toroidal singularities, we produce a modification X0 → X, functorial for toroidal morphisms, making the ideal monomial on a toroidal stack X0. We do this by adapting the methods of [Wło05], 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 → B, aiming to prove functorial semistable reduction theorems.