  We develop a general theory of log spaces, in which one can make sense of the
basic notions of logarithmic geometry, in the sense of Fontaine-Illusie-Kato.
Many of our general constructions with log spaces are new, even in the
algebraic setting. In the differentiable setting, our theory yields a framework
for treating manifolds with corners generalizing recent work of Kottke-Melrose.
We give a treatment of the theory of fans, which are to monoids as schemes are
to rings. By adapting similar results from logarithmic algebraic geometry, we
prove a general result on resolution of toric singularities which can be used
to resolve singularities of a wide class of "log smooth" spaces.
