In the present paper, we introduce bivariate logarithmic weighted spaces of functions and study approximation properties of bivariate exponential sampling series in these spaces. We obtain pointwise and uniform convergence of the series and we determine rate of convergence by introducing a new modulus of continuity for functions belonging to bivariate logarithmic weighted spaces of continuous functions. Furthermore, in order to determine a rate of pointwise convergence, we estimate remainder of Mellin-Taylor formula thanks to the modulus of continuity and we present a quantitative Voronovskaja-type theorem. Finally, we give some graphical representation of approximation of continuous functions by EφWf using kernels which satisfy certain assumptions.