On Sun, Jan 6, 2019 at 5:58 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
Nothing, as far as I can tell. But a logarithm must have a base.
It defaults to base e unless otherwise specified.
Indeed, how do you define log_c(x) if not by log(x) / log(c)...?
I'd define it as the unique continuous function such that f(xy) = f(x) + f(y) and f(c) = 1. You can prove existence as the appropriate multiple of Int_0^x (1/x) dx. e doesn't have to show up unless you want to figure out what the constant you need to multiply by is. As an aside, how do you deal with sentences like the immediately proceeding one, which start with a mathematical expression, which means they can't start with a capital letter, which looks funny to my eye (even if typeset, so that the e is in math-italics). I usually rephrase; in this case, I might have said "The constant e" or "Euler's e" instead of just "e" to start the sentence. Is there some "style guide for mathematicians", and what does it say about starting a sentence with a formula? Andy