Since the natural logarithm, i.e. with base $e$, is very commonly used in research papers and that both $\ln(x)$ and $\log(x)$ are used to denote it, it is natural* to ask which of these notations to use when preparing a paper. The fact that both are used in literature concerning the same topics gives rise to unnecessary confusions and/or definitions. Both have their advantages and disadvantages:

  • The notation $\ln(x)$ bears no ambiguity, as its name is the abbreviation of the French logarithme naturel, or natural logarithm. One does not need to define what it denotes, it is self-explanatory. However, not everyone likes to use it, because...
  • The notation $\log(x)$ is used much more widely for historical reasons as well as notational conventions. However, literally every time it is mentioned in a paper, it is followed by something along the lines of "where $\log(x)$ denotes the natural logarithm, whose base value is $e$", which is not only cumbersome for the reader (who has read this phrase a hundred times before), but can also be avoided by simply using $\ln(x)$.

So which notation is best suited for denoting $\log_e(x)$ and why?

*pun not intended


closed as primarily opinion-based by Gabriel C. Drummond-Cole, Alexandre Eremenko, Chris Godsil, Peter Humphries, Michael Albanese Aug 21 '17 at 13:02

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ My 2 cents: $\log$ for base $e$, $\log_x$ for base $x \neq e$, and $\ln$ never, it is an abomination introduced by the manufacturers of pocket calculators. $\endgroup$ – J.J. Green Aug 21 '17 at 11:20
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    $\begingroup$ At least in some parts of computer science the default meaning of $\log$ is $\log_2.$ For example comparison sorts can't do better then $\log{n!}\approx n\log n$ where $n \log n$ means $n log_2 n.$ So it depends on your audience if you need to spend a sentence saying explicitly what base you intend. $\endgroup$ – Aaron Meyerowitz Aug 21 '17 at 13:35
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    $\begingroup$ There is an implicit convention to use trigraphs rather than digraphs to denote standard functions ($\exp$, $\cos$, $\tan$, $\log$, $\operatorname{det}$, $\lim$, $\sup$, $\operatorname{adj}$, $\operatorname{vol}$, etc.), except in those rare cases in which there is no obvious pronounceable trigraph available (e.g. $\operatorname{tr}$ for the trace, or $\operatorname{st}$ for the standard part of a nonstandard real). Note these are all contractions rather than initialisms. $\operatorname{ln}$ violates these conventions. $\endgroup$ – Terry Tao Aug 21 '17 at 15:13
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    $\begingroup$ One reason to prefer trigraphs over digraphs is that digraphs are far likelier to also occur by accident in one's mathematical expressions, for instance if one is manipulating two variables named $l$ and $n$ then there is some chance of forming the product $ln$ without intending this to be the logarithm. It is far rarer to see three variables $l,o,g$ multiplied together to form $log$. $\endgroup$ – Terry Tao Aug 21 '17 at 15:28
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    $\begingroup$ @J.J. Green: Regarding "it is an abomination introduced by the manufacturers of pocket calculators", $\ln$ was used as early as 1893 in Irving Stringham's book Uniplanar Algebra (see p. 41). You can also find $\ln$ used in Charles Smith's 1900 book Elementary Algebra for the Use of Schools and Colleges (see p. 437). $\endgroup$ – Dave L Renfro Aug 21 '17 at 19:19

In number theory, the notation $ \log $ is commonly used, especially when asymptotics are considered. One also frequently uses the notation $ \log_{k} $ for the $ k $ -th iterate of this function. Indeed the natural logarithm is essentially the only one that matters. This may not be true for other subfields of mathematics.

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    $\begingroup$ You're right for number theory (and complex analysis see, where it is log or Log). $\endgroup$ – Duchamp Gérard H. E. Aug 21 '17 at 11:53

I would divide it into two cases:

  1. If you care about the base being $e$, use $ln$ or state that $log$ is supposed to have base $e$.
  2. If you only want a logarithm but don't care about which one, use $log$ (and maybe state that you don't care about the base). This case often arises when talking about growth of functions, running times of algorithms, etc, as logs with respect to different bases only differ by a constant.

In general, there is no "best solution" for your problem, otherwise there wouldn't be different versions running around. Make sure that your paper gets the message across, is correct and nice to read. If you do everything else right, using $ln$ or $log$ is then simply a question of style.

  • $\begingroup$ Yes, my question concerns exactly case 1: when one cares that the logarithm has base $e$ (i.e. the "natural" logarithm). The whole conundrum is whether to use $ln$, which is not widely used for this purpose currently, or $log$, in which case you always have to specifiy that it has base $e, which, as I stated is cumbersome and annoying. $\endgroup$ – Klangen Aug 21 '17 at 11:10
  • $\begingroup$ Are you sure that $ln$ is not used even when we really want the base to be $e$? I would always go with $ln$ in this case, exactly to avoid this explanation problem, but that is only personal preference. $\endgroup$ – Dirk Aug 21 '17 at 11:17

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