To quote Jaynes, p.91 of PT:TLoS:

In many applications it is convenient to take the logarithm of the odds because of the fact that we can then add up terms. Now we could take the logarithm to any base we please, and this cost the writer some trouble. Our analytic expressions always look neater in terms of natural (base e) logarithms. But back in the 1940s and 1950s when this theory was first developed, we used base 10 logarithms because they were easier to find numerically; the four-figure tables would fit on a single page. Finding a natural logarithm was a tedious process, requiring leafing through enormous old volumes of tables.

Today, thanks to hand calculators, all such tables are obsolete and anyone can find a ten-digit natural logarithm just as easily as a base 10 logarithm. Therefore, we started happily to rewrite this section in terms of the aesthetically prettier natural logarithms. But the result taught us that there is another, even stronger, reason for using base 10 logarithms. Our minds are thoroughly conditioned to the base 10 number system, and base 10 logarithms have an immediate, clear intuitive meaning to all of us. However, we just don't know what to make of a conclusion stated in terms of natural logarithms, until it is translated back into base 10 terms. Therefore, we re-wrote this discussion, reluctantly, back into the old, ugly base 10 convention.

So to answer your question, the only advantage of base e is that "ln" looks tidier than "log10".

Apart from being more intuitively understandable to humans, using base 10 also allows us to multiply by 10 and measure evidence in the familiar unit of decibels.