  We investigate the relationship between stability and the existence of
extremal K\"ahler metrics on certain toric surfaces. In particular, we consider
how log stability depends on weights for toric surfaces whose moment polytope
is a quadrilateral. We introduce a space of symplectic potentials for toric
manifolds, which induces metrics with mixed Poincar\'e type and cone angle
singularities. For quadrilaterals, we give a computable criterion for stability
with 0 weights along two of the edges of the quadrilateral. This in turn
implies the existence of a definite log-stable region for generic
quadrilaterals. This uses constructions due to Apostolov-Calderbank-Gauduchon
and Legendre.
