We study Maulik and Ranganathan's definition of logarithmic stable pairs spaces for toric surfaces relative the toric boundary. We provide a simplified and canonical construction of these spaces.

As a special case we construct the logarithmic linear system: a toric stack whose combinatorics is related to the secondary polytope construction of Gelfand, Kapranov and Zelevinsky. We identify logarithmic stable pairs spaces with the zero set of a section of an explicit vector bundle inside a toric stack, thus providing an explicit description of their virtual fundamental class. The canonicity of our moduli space allow us to perform a number of explicit computations in logarithmic Pandharipande--Thomas theory. These computations give an indication of the range of phenomena in this subject.