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')

It's an interesting debate in notation. Not all log notations show the base, as they mostly assume you know which base you are in (even/especially in the slide rule era, to my understanding, where the base was always assumed by the slide rule in use; which stories posit was fun when mathematicians would default to natural log slide rules and engineers to base-10 log).

This article would be better if showed the base in its notation, but for the most part the bases are determinant in context of each paragraph, and that's roughly par for the slide rule course.

The base is flexible. A slide rule is fundamentally just a set of adding sticks where you perform addition and subtraction in "log space". As long as you convert back out of the log scale it doesn't matter what the base was.

My primary use for logarithms today is verifying how many bits an intermediate calculation is going to need on a calculator without base-2 logs. The base conversion algorithm doesn't care what intermediate base you use as long as they match: bits ~= log2(x) == log10(x) / log10(2) == ln(x) / ln(2).

It's a convention, not a law. If the base is known (or irrelevant) in context (as it is explicitly laid out for several paragraphs), no need to show the base in every expresion.

The article says: "In the world of 2s, log(x) tells you how many times you have to multiply 2 by itself to get x."

In general, log(x) does not refer to one base (although 10 is common). In this case, they specify that log(x) is referring to base 2 in the paragraph above, but yeah it could definitely be more spelled out.