The B-statistic is on the natural log scale (which means base e). In documenting, I followed the usual R practice that logs are always natural unless otherwise stated.
Lonnstedt and Speed (2002) used log10, but natural log is more usual for this sort of statistic in other fields of statistics.
There is no reason why logFC and B should use the same log base as they are quite different sort of quantities. The concept of doubling a log-odds quantity has no particular intuitive meaning whereas it does for a log-fold-change.