This is a joint work with T. Kajiwara and C. Nakayama. Logarithmic abelian varieties are degenerate abelian varieties which live in the world of log geometry of Fontaine-Illusie. They have group structures which do not exist in the usual algebraic geometry. By using the group structures, we give a new formulation of the work of K-W Lan on the toroidal compactification of the moduli space of abelian varieties with PEL structures.