I think it's convention, but it would have been good to explain this a bit more. The previous sentence seems to say "from now on we mean base 2" ("In the world of 2s, log(x) tells you how many times you have to multiply 2 by itself to get x.")

Which base gets to be special depends. Log on its own often means base e; sometimes base 10; and sometimes base 2 (and sometimes 'base whatever, it's not important')