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Wikipedia sez:

The natural logarithm of $x$ is often written "$\ln(x)$", instead of $\log_e(x)$ especially in disciplines where it isn't written "$\log(x)$". However, some mathematicians disapprove of this notation. In his 1985 autobiography, Paul Halmos criticized what he considered the "childish $\ln$ notation," which he said no mathematician had ever used. In fact, the notation was invented by a mathematician, Irving Stringham, professor of mathematics at University of California, Berkeley, in 1893.

Apparently the notation "$\ln$" first appears in Stringham's book Uniplanar algebra: being part I of a propædeutic to the higher mathematical analysis.

But this doesn't explain why "$\ln$" has become so pervasive. I'm pretty sure that most high schools in the US at least still use the notation "$\ln$" today, since all of the calculus students I come into contact with at Berkeley seem to universally use "$\ln$".

How did this happen?

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    It is two less characters =P2010-08-06
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    I've always read it as 'natural logarithm'; in Spanish it works better, though...2010-08-06
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    By "how did this happen?", you seem to be saying there's something wrong, or there's some other alternative notation that is better. Are you? ("log" is already taken by base-10 log in school, and having to write "e" each time is a waste of time.) Here's a notation that's convenient and fills a gap, and it's not at all surprising it's become popular. (Of course, when we grow up we can use log to mean base e or base 2 or whatever is most convenient for us.)2010-08-06
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    @Kevin: +1 for "propædeutic", complete with ligature. I have a pretty good vocabulary (both of my parents were English professors), but this word is a new one for me.2010-08-06
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    (Also +1 because it seems like you have put at least as much scholarship into the question as you are likely to get in an answer.)2010-08-06
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    @Mariano: I've learned "ln" as the latin "logarithmus naturalis". That fits =)2010-08-06
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    A comment: One thing to remember is that not so long ago, using logs (and log tables) was an important way to do practical arithmetic (among engineers, physicists, chemists, etc.). It stands to reason that one would use base 10 for this (if only to make it easy to estimate the logs of various numbers), and so it makes sense to reserve the useful symbol "log" for that case, at least among non-pure mathematicians. A question: what notation did Napier use? Euler? Other 18th and 19th century mathematicians?2010-08-06
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    @Matt: Euler wrote natural logarithms with the letter $l$, e.g., the natural log of 2 was $l2$. There is a link on the page http://eulerarchive.maa.org/pages/E072.html to a copy of the original paper where he gives the Euler product for the zeta-function (Theorem 8) and the very last result, Theorem 19, is the divergence of the sum of reciprocal primes. On the last line of the paper he writes that this series is equal to $l l \infty$, which makes sense since the sum of $1/p$ for $p \leq n$ is asymptotic to $\ln(\ln(n))$ as $n \rightarrow \infty$.2011-06-22
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    @KCd: Dear Keith, Thanks for this useful information and link. Best wishes,2011-06-22
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    "log" is ambiguous. "log10", "log2" and "ln" are not.2013-04-25
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    In Vietnamese we read "logarithm Napier"2016-02-15

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