For a finite dimensional simply-laced simple Lie algebra and an integer , we can attach the logarithmic -algebra . When , is called the triplet -algebra, and studied by many people as one of the most famous examples of -cofinite but irrational vertex operator algebra. However, apart from the triplet -algebra, not much is known about the log -algebras . In this talk, after we construct and their modules geometrically along the preprint of Feigin-Tipunin, first we show the simplicity, -module structure, and character formula of when is in the closure of the fundamental alcove. In particular, for , is simple and decomposed into simple -modules. Second we give a purely -algebraic algorithm to calculate nilpotent elements in the Zhu's -algebra of much easier than straightforward way. Using this algorithm to the cases and , we show that is -cofinite in these cases.