We investigate the distribution of the logarithms, logi, of the currents in percolating resistor networks via the method of series expansions. Exact results in one dimension and expansions to thirteenth order in the bond occupation probability, p, in general dimension, for the moments of this distribution have been generated. We have studied both the moments and cumulants derived therefrom with several extrapolation procedures. The results have been compared with recent predictions for the behavior of the moments and cumulants of this distribution. An extensive comparison between exact results and series of different lengths in one dimension sheds light on many aspects of the analysis of series with logarithmic corrections. The numerical results of the series expansions in higher dimensions are generally consistent with the theoretical predictions. We confirm that the distribution of the logarithms of the currents is unifractal as a function of the logarithm of linear system size, even though the distribution of the currents is multifractal.