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I’ve been working with many students on logarithms and have noted that log has a base of 10 unless specified. Further, I commented that putting a 10 as a subscript to log is redundant, or at least not needed. A student sent me this, from a popular math blog. Now I am wondering if I made a mistake. This image basically implies “when using log, always indicate the base, to avoid confusion.”

enter image description here

Edit: When this came to my attention, I was reminded of a prior question "Why do we write instead of 1 ?" It seems from the replies, this is a different situation.

JTP - Apologise to Monica
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    I think using $\log$ to mean both $\log_{10}$ and $\log_e$ is pretty common. The usual solution is to just announce your convention once the first time you use it in a paper. I would recommend this practice for students as well. – Steven Gubkin Mar 16 '23 at 18:37
  • Is this at university level or earlier? – Tommi Mar 16 '23 at 19:29
  • Thank you. Added a tag, high school level. – JTP - Apologise to Monica Mar 16 '23 at 21:08
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    It seems reasonable to me to match what the calculators say. I think that's least confusing. – Sue VanHattum Mar 17 '23 at 03:12
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    Skimming through the answers here, it seems that the only non-contested log notation is $\ln$ and $\operatorname{lb},$ heheh; for example, $\lg$ appears to, in the answers, variously refer to base 2, base 10, and just an abbreviation of $\log$, which of course has its own multiple interpretations. – ryang Mar 17 '23 at 05:43
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    IMO, use $log10$, $ln$, $log2$, $log_b$ (b is the base) to specify a logarithm and $log$ when the base is irrelevant. – chux - Reinstate Monica Mar 17 '23 at 09:12
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    As with all high-school topics: follow the textbook. Do not introduce your own notation differing from the textbook. NOTE. – Gerald Edgar Mar 17 '23 at 09:23
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    I would also add Computer Scientist thinks $\log_2(50)$ to the slide. If I recall well, I used to denote $\log_2$ with $\text{lg}$, so you might want to add that. For high school I believe it is absolutely necessary to differentiate the cases. However at higher levels, writing the basis is redundant, as the most important property of logarithm is independent of the basis, and it is usually understood from the context (or mentioned once in the beginning of book/article/...). – Kolja Mar 17 '23 at 18:55
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    ISO 31-11 specifies $\mathrm{log}_a$ when the base is explicit, and three abbreviated names $\mathrm{lb}$, $\mathrm{ln}$, and $\mathrm{lg}$ for base 2, $e$, and 10 logarithmss, respectively. I'm not sure who actually uses $\mathrm{lb}$ at all or $\mathrm{lg}$ for base 10. As a computer scientist, I've always used $\mathrm{lg}$ for base 2 (or more precisely, inside Big-O notation where the base is irrelevant). – chepner Mar 17 '23 at 23:04
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    Just one note: In computer science for complexity estimations, it often doesn't matter which basis is used, as there is only a constant factor between them, so a `$\log a$ might just mean "a logarithm of some base, I don't care which one". – Paŭlo Ebermann Mar 19 '23 at 23:41
  • I personally never write "log" on its own. I use "ln" for natural log but will always write an explicit base for log10 and log2. Nice to see others have given this some thought too. The only time I might is, as indicated in Paŭlo's comment, if I find myself in a situation (such as big-O notation) where the base doesn't matter, and all that matters is that it's some logarithm. – Silvio Mayolo Mar 20 '23 at 00:59

5 Answers5

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And a computer scientist thinks that $\log=\log_2$. I am using $\log$ for the natural logarithm by default in all my courses though I clearly state that in the beginning of each course (I teach at the university level). In general, one can use any default base one wishes provided that it has been made clear what it is and that the usage is consistent. At the school level the common tradition is to use $\log$ for base $10$ and $\ln$ for natural logarithm, but if you go higher and look at various textbooks in STEM subjects, both old and modern, this convention is rarely upheld there.

So, my advice is rather to alert the students that $\log$ without base indicated may mean different bases in different contexts and, if the base is not immediately clear, the clarification should be sought and/or requested.

fedja
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    One has to be careful with computer language functions as well... – Jon Custer Mar 16 '23 at 18:54
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    I vaguely recall seeing lg mean "log base 2" in some programing language/library once. "Always clarify avoids confusion" is the right answer. – SkySpiral7 Mar 17 '23 at 21:51
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    As a 40-year computer programmer I've never thought that log=log2. – Barmar Mar 18 '23 at 13:35
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    I've generally seen log-base-2 writte as lg, to distinguish it from log-base-10 log, and log-base-e ln. – supercat Mar 18 '23 at 19:47
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    @supercat - That seems to conflict with this answer from cbeleites (and the comments under that answer), where lg => log-base-10. – Greenonline Mar 19 '23 at 01:41
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    For big-O notation the difference doesn't even matter, since different logarithm bases are a constant factor of each other. – Peter Olson Mar 19 '23 at 05:36
  • @Greenonline: Other notations may be used in Germany, but in the US I've seen lg used as the normal shorthand for log base 2. Of course, "lb" would hardly be the first time standards organizations have decided to coin new terminology for things that people had been defining other ways for decades, such as an uppercase K for 1024. Trying to coin binary-based prefixes doesn't really fix things, given that depending upon context one may need a prefix not only for e.g. 1,000,000 or 1,048,576, but also for 1,024,000 (e.g. a "1.2MB" or "1.44MB floppy" holds 1.2 or 1.44 of the mixed-base unit. – supercat Mar 19 '23 at 06:50
  • @Barmar: You probably didn't learn out of '90s textbooks then. – Joshua Mar 19 '23 at 20:24
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I'm in Germany (chemist, FWIW). I'm familiar with:

  • $\log$: base is unknown/not needed (as in $\log (a) + \log (b) = \log (ab)$

  • natural logarithm: $\ln = \log_e$

  • base 10 logarithm: $\lg = \log_{10}$

In chemistry, both natural and base 10 logarithm are used a lot: The natural logarithm wherever one has to differentiate or integrate (naturally ;-) ), e.g. in kinetics or statistical thermodynamics. In parallel, base 10 logarithm is used a lot in definitions of physical quantities like $\mathrm{pH}$, $\mathrm{pK}_a$, absorbance etc.
In some cases (e.g. Beer-Lambert-law) there is some ambiguity and field-dependent definitions. In that case, we sometimes include the base in the name of the quantity, e.g. the decadic molar absorption coefficent.

  • $\mathrm{lb} = \log_2$ Rare. I've seen it and I recognize it - but I'd spell out this definition rather than assuming any reader to be familiar with this. Don't know whether one could assume computer scientists to be familiar with this.

Update: From this answer on math.sx I learned that this notation is ISO standard.

cbeleites unhappy with SX
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    In CS, we defined $ld = log_2$ (d for dualis) – Luatic Mar 17 '23 at 09:13
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    I'm familiar with lg=log₁₀ (and of course ln=logₑ) in the UK (physics/engineering) usage too. Any others are rare and specified. The times we have to check are calculators and software using "log" without stating the base (Mate calculator on Linux has buttons for ln and log; you can override the base on the latter, but nowhere does it state that the default is 10) – Chris H Mar 17 '23 at 13:56
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    Ah yes lb. For when you want the result to be in pounds. – SkySpiral7 Mar 17 '23 at 21:53
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    $\mathrm{lb}$ is an ISO standard, though that's literally the only reason I know about it. I've never seen it used. – chepner Mar 17 '23 at 23:07
  • (And of course log()+log()=log() is only true if all three logarithms use the same base — so it may be better to specify it e.g. as .) – gidds Mar 17 '23 at 23:20
  • @gidds: In some cases, the people doing this sort of thing are working in Big O notation, or a similar context, where putting the log to any particular base is actually wrong (or at least, sloppy and unnecessary). But that is also a context where writing $2n$ is considered wrong (or sloppy), so probably not something the average student is likely to encounter before college-level computer science. – Kevin Mar 19 '23 at 21:39
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I think stereotypically that log means base 10 and ln means base e. That is how all the 8 different high school and college texts (published 30s to 80s) that I have use it. But I have noticed a (maybe growing?) tendency for other usage. Had to get a formal paper in econ to define their base. (And they were using the log means base e usage, that is more rare...at least for knuckle draggers like me.)

I did a quick Google search on does log mean log base 10 and their were several university math tutor sites still propounding the orthodox religion about common logs. But, a few heretics were seen also. :-(

https://www.google.com/search?q=does+log+mean+log+base+10

"So, when you see log by itself, it means base ten log. When you see ln, it means natural logarithm (we'll define natural logarithms below). In this course only base ten and natural logarithms will be used."

U Minn: http://www.mclph.umn.edu/mathrefresh/logs.html#:~:text=So%2C%20when%20you%20see%20log,natural%20logarithms%20will%20be%20used.

My advice would be to stick with the log means 10 usage since it is still (I think) the most common usage and it's your class that they mostly need it in. I still remember fondly log tables (base 10) being handed out for AP chem into the 90s, in case kids did not have a calculator. ;-)

Maybe briefly mention it to the kids, that some usage is different. (heck "billion" means something different in different countries...and don't get me started on the comma/decimal hassles when passing documents back and forth with Germans).

I wouldn't feel too worried if they see something different in the future. That can be dealt with then by the other teacher. Or if they are really doing sophisticated work in econ or pure math, than they'll be quite capable of dealing with the momentary notation confusion.

Also, FWIW, I wouldn't worry about alternate usage in the more rarified air of higher math courses. Most of your kids need to learn things the old school way and they will be heading for the natural sciences or engineering (which are more traditional). If they end up reading a measure theory text, there's other parts of the text that will vex them more than Napierian being treated as common. ;)

P.s. Of course if you're tutoring and you learn the teach has other usage, defer to him. Or her. Or it.

guest troll
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My experience as university student, researcher and teacher in technical and scientific fields in Italy has taught me that the only widely known unambiguous notation is:

  • ln(x) for base e log (most common)
  • Log(x) for base 10 log (slightly less common – note the capital "L"), but probably to be avoided when writing things by hand, since the capitals can be easily be warped by quick handwriting.

The notation log(x) (sometimes abbreviated as lg(x) in Italy) is used to indicate mostly natural logarithm (especially in math courses), but also frequently base 10 logarithm (in some engineering courses, for example). Some computer scientists use it to indicate base 2 log, but it is far less common (but not at all rare).

What my math professors at university always said was to check my assumptions when using the "baseless" notation and to check any paper/book that used that notation for disambiguation clues, such as symbol lists, because there was no universal standard or practice to rely on.

The only important thing is to be consistent: if you choose to use "log" for natural logarithms, then state so upfront, remind that choice often to your students (especially at the beginning), point out that it is not an universal choice, and never, ever, use that symbol for something else in the same context or with the same group of students.

My usual choice nowadays (in technical fields) is to use log for base 10 (so I don't have capitalization issues) and ln for base e, so it matches any pocket calculator a student may be using.

LorenzoDonati4Ukraine-OnStrike
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Partially tongue in cheek, but it would feed two birds with one apple :

"Henceforth this book | article | thesis uses the following notation

$ ln( \sqrt {-1} ) = \frac \pi 2 i $

(*) or

$ log_e( \sqrt {-1} ) = \frac \pi 2 i $

(*) or

$ Log_{2}( 1 T_i ) = 40 $

(*) or

$ log( \sqrt {-1} ) = \frac \pi 2 j $

(*) or

$ log( \sqrt {-1} ) = \frac \pi 2 i $ "

(*) - only one used

Then we'd know if we're in the good company of mathematicians, scientists, computer scientists, electrical engineers or all other engineers.

Nick Myra
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