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Question: How and why did the discipline of mathematics come to use $\log(x)$ and not $\ln(x)$ in its writing - papers, books, teaching?

Clearly $\ln(x)$ is unambiguously $\log_e(x)$, but $\log(x)$ is ambiguous.

Almost all textbooks, papers and informal formats like blogs and YouTube videos use $\log(x)$ and assume base $e$.

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    $\begingroup$ The notation "ln" is a recent innovation. Mathematicians used to (and most still do) write "log" for the natural logarithm. "ln" first used 1893, "log" first used 1647, but "Log." used 1624. See mathshistory.st-andrews.ac.uk/Miller/mathsym/functions $\endgroup$ May 1, 2022 at 14:32
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    $\begingroup$ I understand that use of $\log$ instead of $\ln$ may be a sign of academic snobbishness. A sniffy-nosed colleague of mine stated categorically that using $\ln$ was unprofessional and showed that its user was unsophisticated and had obviously never risen further than a mediocre grade at school. As a consequence I use $\ln$ universally. $\endgroup$ May 6, 2022 at 9:02
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    $\begingroup$ Is it really ``the discipline of mathematics"? It seems to me the use of ln versus log is a cultural thing, with log dominant in the US. In my university education in Eastern Europe at the end of the last century, ln for natural logarithm was alive and kicking. $\endgroup$ May 9, 2022 at 1:05
  • $\begingroup$ @MargaretFriedland the custom in Eastern Europe is more mixed now. In the last 10+ years in Russia I often saw the notation ln used in writing at the blackboard (a culture shock the first time, coming from the US), but in some recently published pure math books there I have found log. I wonder if a more common habit of writing papers in English has made log in place of ln more accepted. $\endgroup$
    – KCd
    Dec 13, 2022 at 6:42

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For me, handwritten especially, but also typeset, "ln" is easier to mis-read than "log". And since I do not ever use any logarithm but "base $e$", (except, rarely, base $2$ for some information-theory things maybe), there is simply no ambiguity.

It's true, it is easier to start kids with logs base $10$, and in calculus make the distinction to/from "ln", ... but that is a transitional phase, I think.

(Financial computations do also use logs base $e$, for reasons of natural phenomena! But, one way or another, almost no people have any reason to directly worry about which logarithm was intended...)

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  • $\begingroup$ Thanks Paul. Can I ask how typeset "ln" can be misread? What are the other possibilities for interpretation? $\endgroup$
    – Penelope
    May 1, 2022 at 0:02
  • $\begingroup$ @Penelope, well, the "l" in the "ln" may suggest that it's a capital "i", or a numeral "one". Sure, a little concentration can usually resolve that... but there's nothing at all similar to "log"... When scanning things, ease of such details makes a difference, for me. Maybe not for others. :) $\endgroup$ May 1, 2022 at 3:44
  • $\begingroup$ @paulgarrett In handwriting, if one uses joined-up writing, it is difficult to mistake $\text l$ for $\text I$. Unfortunately the concern is more about the universal use of sans-serif fonts in computers, coupled with a careless ignorance about using professional typesetting software that allow one to mistake $\ln$ with $\text {In}$. Seriously, whose stupid idea was it to standardise on Arial instead of something with serifs like TNR? They deserve to be kicked around the world by the backside. :-) $\endgroup$ May 6, 2022 at 9:08
  • $\begingroup$ @PrimeMover, yes, in any sort of cursive, the (in TeX) $\ell$ is easily distinguishable. But, in modern times wherein many people never use cursive, block-printing or stick-printing leaves "ell" as just a vertical slash... and "one" is a vertical slash... and upper-case "i" is a vertical slash... and an uncrossed "t"... :) $\endgroup$ May 6, 2022 at 18:01
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    $\begingroup$ @paulgarrett Bah. the youth of today. No respect for their elders. When I was a young man we were expected to communicate clearly. Just a minute, there's someone on my lawn. "Oy! Gerroff ..." $\endgroup$ May 6, 2022 at 18:14

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