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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 =P – Casebash Aug 6 '10 at 6:03
I've always read it as 'natural logarithm'; in Spanish it works better, though... – Mariano Suárez-Alvarez Aug 6 '10 at 6:11
@Mariano: I've learned "ln" as the latin "logarithmus naturalis". That fits =) – Jens Aug 6 '10 at 9:28
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? – Matt E Aug 6 '10 at 10:27
@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 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 $n \rightarrow \infty$. – KCd Jun 22 '11 at 20:46

As noted in the original question, Wikipedia claims that the ln notation was invented by Stringham in 1893. I have seen this claim in other places as well. However, I recently came across an earlier reference. In 1875, in his book Lehrbuch der Mathematik, Anton Steinhauser suggested denoting the natural logarithm of a number a by "log. nat. a (spoken: logarithmus naturalis a) or ln. a" (p. 277). This lends support to the theory that "ln" stands for "logarithmus naturalis."

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In my old school days, which I underwent in the former GDR, I learned that $\ln(x)$, $\operatorname{ld}(x)$, and $\operatorname{lg}(x)$ originate from the names logarithmus naturalis, logarithmus dualis, and logarithmus generalis that were used in the ancient mathematical publications written in latin. – Björn Friedrich Dec 7 '15 at 20:55

Cajori, in his History of mathematical notations, Vol. II, as far as I can see, mentions the notation «$\operatorname{ln}$» once when he is talking about notations for logarithm. He refers to [Irving Stringham, Uniplanar Algebra, (San Francisco 1893), p. xiii] as someone who used that notation; the way this guy is mentioned makes me doubt he was the first or alone in this, though (I'd love to know what 'uniplanar algebra' is (was?)!)

The mention of «$\operatorname{ln}$» is quite minor, and I would guess that at the time Cajori was writing (the volume was completed in August 1925) the notation was essentially not used, for otherwise he would have been more interested in it.

PS: I think I got the link from someone here, or on MO... is pretty relevant!

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Here's an actual image of the use of ln on the work in question: – Jason Dyer Aug 6 '10 at 14:31

It is probably at least in part an instance of consumer lock-in, the phenomenon where the advantages of the majority of the market using the same or compatible products outweigh sufficiently minor differences of one version of the product versus another. (When I learned this term, the canonical example was VHS versus Beta. I am just about old enough to remember that when VCR terminology first came out, vendors would carry both. Already when the first video stores opened up there was more product available in VHS, so a lot of stores would have just one rack with all the Beta products. And eventually of course Beta died. However those who remember and care seem to largely agree that Beta was the superior technology. Apologies for giving such an old-fogey example. Can someone suggest something more current?)

In particular, when it comes to electronic calculators, having everyone agree what's going to happen when you press a certain button is a good thing. (In fact, another lock-in phenomenon is that when I was in high school in the early 90's, most students had Texas Instruments calculators of one kind or other. Among the real geeks it was known that the "cadillac of calculators" was actually the Hewlett-Packard, which used reverse polish notation. Serious CS people appreciate RPN, but the problem is that if you're a high school kid and pick up such a calculator for the first time, it's very hard to figure out what's going on. I haven't seen an HP calculator for many years.) The notation $\ln$ is simple and unambiguous: you don't have to like it (and I don't, especially), but you know what it means, and it's easier to fit on a small calculator key than $\log_e$. I think if you're first learning about logarithms, then base ten is probably the simplest (to have any clue what $e$ is other than "about $2.71828...$" requires some calculus, and is in my experience one of the more subtle concepts of first year calculus), so it's reasonable to have that be the standard base for the logarithm for a general audience.

Also, I'm sure everyone here knows this but I wish my calculus students had a better appreciation of it: exactly what base you take for the logarithm only matters up to a multiplicative constant anyway, since $\log_c x = \frac{\log_a x}{\log_a c}$. So it's no big deal either way.

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The HD-DVD vs. Blue-Ray war is a more modern example, I guess. – Mariano Suárez-Alvarez Aug 6 '10 at 8:41
So who won that war? (Yes, I am little out of touch on this stuff. Once I almost bought a movie on Blue-Ray by mistake...) – Pete L. Clark Aug 6 '10 at 10:23
It is possible to get at e in a meaningful way in a typical advanced algebra/precalculus course in high school from the idea of compound interest using smaller and smaller compounding intervals (an informal limit), but it's certainly not as important/memorable a property of e as its calculus properties. – Isaac Aug 6 '10 at 13:45
@Pete: Blu-ray won. HD-DVD is now only used by Xbox 360 (for backwards compatibility reasons). Blu-ray will probably replace DVD within the next 5~10 years. – BlueRaja - Danny Pflughoeft Aug 6 '10 at 16:22
Though this is offtopic: The HD-DVD vs Blu-Ray is not an example of the same thing, because it's not clear that HD-DVD was any better. Rather, Sony learned from the mistakes of losing over Betamax, and made sure it didn't happen to their Blu-Ray this time. An (often-cited) example would be the QWERTY v/s Dvorak keyboard layout issue, where QWERTY is the standard layout despite Dvorak being more efficient and easier on one's fingers: the gains are minor for most people, and not worth it. – ShreevatsaR Aug 6 '10 at 17:04

I suggest that everybody from now on uses ln, lg, and lb respectively for the natural, decimal, and binary logarithmic functions, reserving log for when the base is displayed explicitly or is some fixed arbitrary number if the scale doesn't matter. I do (with explanation if necessary at the first instance) . What could be simpler?

This convention is the ISO standard (since 1992); see Wikipedia.

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lg is too similar to log but lx (the roman numeral "x") seems workable. ;) – Heath Hunnicutt Aug 6 '10 at 20:36
Someone else has had the same idea. To what extent it has caught on is debatable. I, for one, know plenty of people who use lg as the base 2 logarithm. – Larry Wang Aug 7 '10 at 15:38
@KH: Thanks for the reference. Isn't it a good idea to use ISO standards? They seem pretty carefully thought out to me. – John Bentin Aug 7 '10 at 20:32
In theory, sure, but for better or worse, things like "what are my colleagues used to" or "what is most convenient for me" often displace "what the ISO thinks" on people's lists of things to consider when deciding what nomenclature to use. – Larry Wang Aug 8 '10 at 12:42
@KH: Which is to be master---Humpty Dumpty or the ISO? Time will tell, but my money's on the ISO in this case. – John Bentin Aug 8 '10 at 14:04

I don't have enough rep. on this site to post a comment on Gerard's answer, so I'll write this as an answer. Those who say "tan" is more common than tg should keep in mind where they live. In Russian the standard abbreviation is tg, not tan. They write sin and cos, but the trig functions tan, cot, and csc have alternate abbreviations. See the first line atТригонометрические_функции, for instance, where the trigonometric functions are named and then abbreviated in parentheses.

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Off-topic, but you have reminded me about one of the things that annoyed me about using either Jahnke-Emde or Bronshtein: they keep using "tg" instead of "tan". Still, it is a minor quibble with respect to the wealth of information in those books. – J. M. Aug 8 '10 at 10:32
Off-off-topic, but many pages from Jahnke-Emde (I loved that book in high school) are used in the movie "Pi". The movie also has the formula for the golden ratio, but they got it wrong: they wrote "a/b = (a+b)/b". I put an entry into tv-tropes about this:) – marty cohen Aug 27 '11 at 2:30
In China, maybe following USSR, old textbooks use $\operatorname{tg}$ and $\operatorname{ctg}$, but now they're replaced by $\tan$ and $\cot$. – Frank Science Jul 12 at 2:10

One simple possible reason

It is very difficult to come up with a notation that is concise, correct and understandable by at least two people. - P A M Dirac / Richard Feynmann

This one is concise, correct and is understandable at least by you and me!

Generally, notations used in popular textbooks spread like virus. Books become popular because of good notation and notation becomes widespread because book is popular.

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I think the most difficult part of the claim to verify is the correctness. – BBischof Aug 6 '10 at 14:01

Well, we call $e^x$ natural exponentation and therefore \log_e(x) natural logarithm. This one is abbreviated as \ln (for logarithmus naturalis).

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The log abbreviation was more understandable then the lg abbreviation, and also was more natural compared to the sin, cos, tan abbreviations. Compare the tg notation versus tan. They are both in use, however tan is more common.

In the case of log_a(x) we have two important special cases: log_10 and log_e. Log_10 was so common, especially in high school, that it became abbreviated to log.

Log_e is/was more difficult to understand at high school level - I can remember that. The notation ln to distinguish this special case is helpful especially for beginners. Two different set of rules for two different scenarios (log=log_10 and ln=log_e). Later on the can perhaps understand the general case.

To add to the confusion: my teachers at the university used the notation log to mean ln!! From there point of view any extra notation should be avoided and log_10 had no particular importance to them.

So the ln notation turns out to be useful from a didactical perspective.

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For beginners, I think explicitly writing out $\log_e$ is easier to understand. ln is just a notational shorthand. – Mechanical snail Aug 26 '11 at 21:55

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