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Recently, I took a higher level mathematics exam. When I took it, I held the notion that $log(x)$ was assumed to be base ten and natural logarithms were assumed to be $e$. Come to find out in the directions that $log(x)$ is assumed to have base $e$. I had seen some people use this notation in general mathematics when studying how to raise complex numbers to complex powers but to be honest other than that very rarely, especially with generic integrals.

$\textbf{Question}:$ My question is when is the turning point in college/time from logarithms, i.e. $log(x)$, having base ten to all of the sudden base $e$?

$\bullet$ I mean are we going to sometime soon start using this notation with AP Calculus where natural logarithms are always used?

<p><span class="math-container">$\bullet$</span> Are people seeing this notation in Calculus I/Calculus II at other schools? If so, I would be curious as to what schools commonly use this notation.</p>

I see this switch in notation to be a $\textbf{major issue}$ in conflation of symbols which is why I am posting this question.

W. G.
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    This is probably culture dependent - I personally use $\ln x$, $\operatorname{lg} x$, $\operatorname{ld} x$ for base $e$, base $10$, base $2$ -- and $\log$ only when constants don't really matter (i.e., big Oh stuff) – Hagen von Eitzen Jan 27 '19 at 14:58
  • That's a good point! – W. G. Jan 27 '19 at 15:01

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There are three especially popular bases for logarithms, $2,\,e,\,10$. (Occasionally something like the golden ratio is useful instead, but you won't need to consider these unless they're very explicit about it.) As a rule, logarithms are base-$2$ in a information-theoretic context concerned with binary computing, base-$e$ if calculus comes up in the question, or base-$10$ if you need to talk about large or small numbers concisely. Given all of that, the only time you won't know which is needed to handle a question is when you're asked to compute a logarithm, presumably to some specific number of decimal places or significant figures; and even then you may as well just say "here's $\ln x$ and here's $\log_{10} x$".

J.G.
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