What is the convention for log notation in biology?
There is a reason for prevalent usage of log2 and log10 compared to loge in biology and other experimental sciences. Usually while doing measurements we are generally interested in fold changes and we generally talk in the sense of two-fold (or multiples of two) or ten-fold.

Doubling is a common phenomenon at least in case of growth and so a two-fold change would represent an event of doubling. log2 gives you the extent of "doubling". Even in computer systems memory is always in the exponents of two (because of binary); so even there log2 would make sense.

In biology you would generally not see a ten-fold change but in physics it is relatively common. Most values are represented in scientific notation which basically uses an exponent of 10. Also the metric units are in multiples of 10. Here log10 would be more suitable.

loge has a special mathematical importance i.e. it can be denoted as a series. The euler's number has other important properties too. Therefore in most theoretical studies, both in biology and physics, loge is preferred because it makes more mathematical sense.

To sum up, experimentalists prefer log2 or log10 because they are better for representing measurements whereas theoreticians prefer loge because of its mathematical importance.

W.r.t the old paper that reports log-log relationships, the base of the logarithm wouldn't make any difference to the slope. But most likely it must be log10 (I think back then log10 was the default log; log2 would be much more suitable for lengths and weights of organisms compared to log10 because these measures would not vary in multiples of 10).