  We define the categories of log motives and log mixed motives. The latter
gives a new formulation for the category of mixed motives. We prove that the
former is a semisimple abelian category if and only if the numerical
equivalence and homological equivalence coincide, and that it is also
equivalent to that the latter is a Tannakian category. We discuss various
realizations, formulate Tate and Hodge conjectures, and verify them in curve
case.
