  We introduce a natural geometric framework for the study of logarithmically
divergent integrals on manifolds with corners and algebraic varieties, using
the techniques of logarithmic geometry. Key to the construction is a new notion
of morphism in logarithmic geometry itself, which allows us to interpret the
ubiquitous rule of thumb "$\lim_{\epsilon\to 0} \log \epsilon := 0$" as the
natural restriction to a submanifold. Via a version of de Rham's theorem with
logarithmic divergences, we obtain a functorial characterization of the
classical theory of "regularized integration": it is the unique way to extend
the ordinary integral to the logarithmically divergent context, while
respecting the basic laws of calculus (change of variables, Fubini's theorem,
and Stokes' formula.)
