  We develop a theory of motives with compact support for logarithmic schemes
over a field. Starting from the notion of finite logarithmic correspondences
with compact support, we define the logarithmic motive with compact support
analogous to the classical case for schemes. We then establish an analog of a
Gysin sequence and, assuming resolution of singularities, a K{\"u}nneth
formula. This implies that our theory is $\overline{\square}$-invariant, which
presents a critical feature that is absent in the classical case. Further
assuming resolution of singularities, we prove a duality theorem for log
schemes which we use to establish a cancellation theorem for log schemes whose
underlying scheme is proper. Moreover, we discuss new homology and cohomology
theories for log smooth fs logarithmic schemes based on our results.
