The natural logarithm is often represented by several different notations:

  • $\log_e x$
  • $\log x$ (although this is also used for logarithms with a base of 10)
  • $\ln x$

It is the third notation that has me wondering. Why is $\ln$ used, and not, say, $\text{nl}$? My two theories about this are

  1. It is an abbreviation for "natural logarithm" in a non-English language
  2. It is meant to correspond with the "$l$" in typical logarithmic notation.

Why is the notation $\ln$?

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    $\begingroup$ I guess this just stands for Logarithm Natural (in some languages like French, they use this order of the words). Do you really think that this is really an interesting/important question on the history of Math and Sciences? $\endgroup$ – Alexandre Eremenko Sep 7 '15 at 18:24
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    $\begingroup$ @AlexandreEremenko It's about the history of notation, which is a valid subject. $\endgroup$ – HDE 226868 Sep 7 '15 at 18:25
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    $\begingroup$ In pure mathematics (unlike in applied computations) they use only natural logarithm. So they abbreviate it as $\log$. $\endgroup$ – Alexandre Eremenko Sep 7 '15 at 18:26
  • $\begingroup$ @AlexandreEremenko Yep, I read that; $\ln$ is used mainly in physics and engineering. $\endgroup$ – HDE 226868 Sep 7 '15 at 18:26
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    $\begingroup$ Formally this is a valid subject. But on my opinion the question is trivial and not important. $\endgroup$ – Alexandre Eremenko Sep 7 '15 at 18:28

As far as I am aware, the mathematical operator commonly spelled $\ln$ is an abbreviation of the Latin term, logarithmus nātūrālis.

I am not sure who first used this abbreviation, but I suppose it very well may have been Napier.

I remember having seen ln written as an abbreviation of two words in the form $l.n.$

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    $\begingroup$ This question (of course) has been asked before. See math.stackexchange.com/questions/1694/…, and note the answer by Velleman, which gives support to this answer. $\endgroup$ – KCd Sep 10 '15 at 1:44

According to Earliest Uses of Function Symbols :

$\ln$ (for natural logarithm) was used in 1893 by Irving Stringham (1847-1909) in Uniplanar Algebra (Cajori vol. 2, page 107).

Thnaks to KCd's reference in his comment, we have an earlier occurrence :

  • $\begingroup$ I should have seen that! I was looking in a related page the other day. Any idea why Stringham chose it? $\endgroup$ – HDE 226868 Sep 7 '15 at 18:49

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