You means that it is not a good idea to solve this problem by the method of logarithmic transformation? Except for choosing finer meshes and the method of lograrithmic transformation, it seems that there is not anything other strategy to solve this kind of problem.
Yes, I think the logarithmic transformation is unlikely to help. So use it only as a last, desperate resort. The meshing is much more important, probably the most important setting. Other settings that, in my experience are worth playing around with, are: the tuning parameters for the various stabilization techniques offered by the Transport of Diluted Species interface, the (relative/absolute) tolerances, and (manual) scaling of dependent variables — roughly in this order of importance.