# When writing log, do you indicate the base, even when 10?

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.”

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

• 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. Mar 16 at 18:37
• It seems reasonable to me to match what the calculators say. I think that's least confusing. Mar 17 at 3:12
• IMO, use $log10$, $ln$, $log2$, $log_b$ (b is the base) to specify a logarithm and $log$ when the base is irrelevant. Mar 17 at 9:12
• As with all high-school topics: follow the textbook. Do not introduce your own notation differing from the textbook. NOTE. Mar 17 at 9:23
• 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/...). Mar 17 at 18:55

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.

• One has to be careful with computer language functions as well... Mar 16 at 18:54
• I vaguely recall seeing lg mean "log base 2" in some programing language/library once. "Always clarify avoids confusion" is the right answer. Mar 17 at 21:51
• As a 40-year computer programmer I've never thought that log=log<sub>2</sub>. Mar 18 at 13:35
• I've generally seen log-base-2 writte as lg, to distinguish it from log-base-10 log, and log-base-e ln. Mar 18 at 19:47
• @supercat - That seems to conflict with this answer from cbeleites (and the comments under that answer), where lg => log-base-10. Mar 19 at 1:41

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.

• In CS, we defined $ld = log_2$ (d for dualis) Mar 17 at 9:13
• 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) Mar 17 at 13:56
• Ah yes lb. For when you want the result to be in pounds. Mar 17 at 21:53
• $\mathrm{lb}$ is an ISO standard, though that's literally the only reason I know about it. I've never seen it used. Mar 17 at 23:07
• (And of course log(𝑎)+log(𝑏)=log(𝑎𝑏) is only true if all three logarithms use the same base — so it may be better to specify it e.g. as 𝑛.) Mar 17 at 23:20

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. :-(

"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."

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.

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.

• Would you mind editing your answer to make it less tongue-in-cheek? That'd make it more useful to anyone reading it. Mar 19 at 19:30

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.

• Gosh, $\ln(x)$ means base $2,$ $\lg(x)$ means base $e,$ and the capital 'L' in $\operatorname{Log}(x)$ signifies base $10 ?$ These are indeed far removed from all corners of mathematical practice. Mar 19 at 14:29
• and in other contexts, "Log(z)" with the uppercase L is used for the main branch of the complex logarithm, where the general log(z) would be multi-valued... Mar 19 at 18:43
• @ryang Sorry, I didn't recheck my post. I a dumb typo crept in at the beginning (as my last sentence could have hinted). See my correction. Mar 20 at 11:03
• Haha ok, noted. That’s perhaps why your post got downvoted (not from me though). -( Mar 20 at 11:06
• @ryang BTW, thanks for the comment. Without it the post could have been rightfully downvoted to oblivion! :-) Mar 20 at 11:10