Exactness, integrality, and log modifications

In this paper we discuss log blow-up’s, introduced by Kazuya Kato, and define the concept of log modifications. Using this concept we prove that any morphism f: X —> Y of locally noetherian fs log schemes with underlying structures of f and Y quasi-compact can be modified to an exact morphism, and moreover to an integral morphism. By a well-known fact on the underlying structure of an integral morphism this result can be considered as a weak log-version of flattening theorem by Raynaud and Gruson.