Building upon recent work by Binda, Park, and Østvær we construct a theory of motives with compact support in the setting of logarithmic algebraic geometry. Starting from the notion of finite logarithmic correspondences with compact support we define the logarithmic motive with compact support analogous to the classical case. After establishing a Gysin sequence, we prove a Künneth formula, which as a special case, proves homotopy invariance of the logarithmic motive with compact support. This presents an important distinction from the theory of motives with compact support which is not homotopy invariant. Relating our theory to the classical theory we provide an affirmative answer to a question raised in Binda--Park--Østvær concerning the theory's relation to the classical theory. We then prove an analogue of the classical duality theorem, which together with a calculation of the logarithmic motive with compact support of the affine line, culminates in a proof of a cancellation theorem for logarithmic schemes. Moreover, we provide a new homology and cohomology theory for logarithmic schemes, and give a new homotopy invariant generalization of Bloch's higher Chow groups to logarithmic smooth fs logarithmic schemes.