The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L^2(\mathbb R) maximum principle, in the form of a new "log'' conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance \|f\|_1 \le 1/5 . Previous results of this sort used a small constant \epsilon \ll1 which was not explicit. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy \|f_0\|_{L^\infty}<\infty and \|\partial_x f_0\|_{L^\infty}<1 . We take advantage of the fact that the bound \|\partial_x f_0\|_{L^\infty}<1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.