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.)