Gareth's post is very well written. As I see it, the only problem is that "being in the same community" is not an equivalence relation — so there are overlapping groups that tend to use the same notation for different things. I almost always use ln when I mean natural log, just because it's unambiguous. I'm not sure about other countries (though googling had shown that many Europeans use ln). But as for the U.S.: I don't see any good argument *against* everyone's using ln to mean natural log. Virtually everyone who knows what natural log means has seen that notation when they took calculus. And that will avoid any confusion between log_10 and log_e. Who knows how many bridges already fell down for just this reason? --Dan P.S. And while we're at it, let's standardize the meaning of "natural numbers" and the symbol N for them — or else come up with new notation that is unambiguous. Gareth wrote: ----- "ln" is no more immediately obvious than "log"; in either case you need to know what the notation means to do anything useful with it. The same goes for every other bit of mathematical notation there is. What's "secret" about it? Only the fact that in some other communities other than that of pure mathematicians it happens that "log" is used to mean something else. That isn't the result of any kind of in-crowd status-symbol psychological bullshit. It's just that if you happen to need one kind of logarithm much more often than others it's natural (ha!) to use "log" to denote that; and it happens that for pure mathematicians that happened with the natural log, while for engineers and schoolteachers it happened with the base-10 log. Natural logs are still much the most, er, natural kind in pure mathematics (though maybe base-2 logs are more important now than they were 50 years ago). There's less excuse for the dominance of base-10 logs in engineering and school teaching, now that slide rules and log tables have been so thoroughly supplanted by other means of computation. But of course the real reason why the notation persists in both cases is simply tradition: changing would require lots of people to abandon the notation they're used to for very little benefit, and invalidate lots of existing textbooks, journal articles, etc. It might be no bad thing if pure mathematicians and engineers and schoolteachers all got together and agreed never again to use unqualified "log" to denote any particular base, and to use (say) "lg" for base 2, "ln" for base e, and "ld" for base 10. Realistically, though, it's not going to happen. (Also, "ln" is distractingly hard to read because lowercase-l looks like capital-I and digit-1 and so forth and needs all the extra context it can get, and "ln" is harder to say than "log"; that agreement would be pretty much a pure loss for the pure mathematicians.) Among people who aren't pure mathematicians it is common to write angles in degrees, and common -- albeit sloppy -- to leave off the degree signs. Is it in-crowd status-symbol psychological bullshit for mathematicians to define the sine function so that its period is 2pi rather than 360? Among people who aren't pure mathematicians it is common to use terms like "compact", "real", "manifold", "similar", etc., with meanings very different from those pure mathematicians give them. Is it in-crowd status symbol psychological bullshit for pure mathematicians to use their meanings for these words? Of course not. Same for "log". -----