# What is the accepted notation for natural logarithms, trig functions and powers?

Actually, I have several different questions:

• What is the accepted notation at this site for the natural logarithm: $\log x$ or $\ln x$?
• Is it OK to write $\sin 2\pi x$ instead of $\sin(2\pi x)$? $\log 2\pi$ instead of $\log(2\pi)$?
• Is it OK to write $\sin^n x$ instead of $(\sin x)^n$?
• What about special functions? Are these OK: $J_\nu^2(x)$, $\text{Li}_2^2\frac1{e^\pi}$, $\text{Si}^3 x$
• This should probably have been a question on the main site, no? – Mariano Suárez-Álvarez May 21 '13 at 17:05
• @MarianoSuárez-Alvarez: since the question specifically asks about what is the accepted notation on this site, I think this is fine on meta. – robjohn May 21 '13 at 17:51
• On most scientific calculators, $\log$ means base 10 and $\ln$ means base $e$. – Joel Reyes Noche May 21 '13 at 23:08
• I think that all your suggestions are fine. We should be allowed to trust readers to interpret the meaning taking the context into account. If I talk about the signal $\cos2\pi f t$ or about the sine wave $\sin\omega t$, I teach/expect my students to insert parens at the correct place. I warn them against using $\sin\omega t+\varphi$ in place of $\sin(\omega t+\varphi)$ when dealing with phase shifts, though. Not because it would not be hard to interpret that also from the context, but because they are likely to fool themselves. Invariably still happens :-( – Jyrki Lahtonen May 23 '13 at 8:20

My (slightly opinionated) observations:

• $\ln x$ is safe to use.
• $\log x$ is almost always safe to use as $\ln x$, except in the context of secondary school mathematics (some school systems use $\log$ for base $10$ logarithms).
• $\lg x$ should be avoided: some people use it as $\log_{10}$, others as $\log_2$. (Logarithm notation from Wikipedia, Lg article from Wolfram).
• For trigonometric functions, it is better to use the notation in the right column. (Both because it's more common here, and because it's native to MathJax): \begin{align*} \operatorname{tg}x &= \tan x\\ \operatorname{ctg}x &= \cot x \\ \operatorname{sh}x &= \sinh x \\ \operatorname{ch}x &= \cosh x \\ \operatorname{th}x &= \tanh x \\ \operatorname{cth}x &= \coth x \end{align*}
• Inverse trigonometric functions can be written as either $\sin^{-1}$ or $\arcsin$. Although I prefer the latter personally (less ambiguous), there are also reasons to use the former: arc-notation for inverse hyperbolic functions, e.g., $\arctanh$, is not native to MathJax
• For positive powers of trig functions, $\sin^{n}x$ is fine: $\sin^2 x$ looks better than $(\sin x)^2$. For negative powers, one should use parentheses to avoid confusion, with $\sin^{-1}x$ being $\arcsin x$.
• For binomial coefficients, the notation on the right is preferable. $$C^m_n ={}_nC_m = \binom{n}{m}$$
• For non-elementary functions (perhaps with exception of the well-known $\Gamma$ and $\zeta$): a short remark at the end of post would be nice, unless the context makes the meaning absolutely clear.

For example,

• $\operatorname{Li}$ is the integral logarithm, with the convention $\operatorname{Li}(2)=0$.
• $B_n$ are Bernoulli numbers with the convention $B_1=1/2$.
• Notes: 1. When not dealing with natural logarithms, play it safe and always indicate the base. 2. I've found that Russians vastly prefer the notation on the left column for trig/hyperbolic functions. 3. $\binom{n}{k}$ is also sometimes denoted as ${}_n C_k$ in elementary texts. 4. (Analytic) Number theorists use the version of the logarithmic integral that is zero at $2$; most other people prefer the version that requires the use of a principal value integral. (cont'd) – J. M. isn't a mathematician May 21 '13 at 19:03
• 5. On the subject of what $B_1$'s sign really ought to be, this is a nice page on the matter. – J. M. isn't a mathematician May 21 '13 at 19:04
• @J.M. I know that notation on the left is standard in Russian textbooks. My point is that for the purpose of posting on Math.SE, the notation on the right is preferable, as more likely to be understood. – 75064 May 21 '13 at 19:11
• @J.M. I edited to include your example of binomial notation. Also made it clear that two lines at the end are an example of what a notational remark could be, not my opinion on the "right" notation for those things. – 75064 May 21 '13 at 19:19
• BTW: for inverse hyperbolic functions, I believe the preferred notation is $\mathrm{artanh}$, not $\mathrm{arctanh}$. – J. M. isn't a mathematician May 22 '13 at 2:16
• @J.M. oh, huh. I've never known that... now I feel silly for writing $\mathrm{arccosh}$ all over the place. – Willie Wong May 22 '13 at 7:14
• @Willie, I think it was a question on main. (No (circular) arcs involved, so don't use the $\mathrm{arc}$- prefix) I remember Arturo mentioning that he himself was accustomed to $\mathrm{arg\,cosh}$. – J. M. isn't a mathematician May 22 '13 at 7:20
• Ah, found it. – J. M. isn't a mathematician May 22 '13 at 7:26
• On the other hand Maple uses $\operatorname{arctanh}$ for the inverse of $\tanh$. – Américo Tavares May 22 '13 at 19:58
• Mathematica also uses ArcSinh and its cousins. I recall being very surprised by this the first time I saw it given that my undergraduate texts were very clear about it being arsinh (going back to latin area sinus hyperbolicus). Is adding that extra 'c' a North American thing? My limited experience suggests that it might be, but I would not call the evidence I have statistically significant. – Jyrki Lahtonen May 23 '13 at 8:13
• The ISO notation for binary and decimal logarithms is ... odd, to say the least, and not, in my experience, very common. In particular, I’m skeptical of the claim that most would understand $\lg$ as $\log_{10}$: I’ve almost never seen it used that way. I recommend using $\lg$ for $\log_2$, though it probably warrants a comment. – Brian M. Scott May 29 '13 at 7:03
• @J.M. - Observation about the arc: in Dutch, the inverse trig functions have names starting with boog, which means arc (a piece of a circle). I don't know why. Wikipedia has a paragraph on the etymology of arc. en.wikipedia.org/wiki/… – Daan Michiels May 29 '13 at 7:40
• @Daan, I don't dispute the $\mathrm{arc}$- prefix for inverse trigonometric functions; as a matter of course, I vastly prefer to write $\arctan$ instead of $\tan^{(-1)}$; it is the inverse hyperbolic functions that seem to be murky. I have merely said that the convention of writing $\mathrm{artanh}$ is what I am accustomed to, and I happen to agree with the justification why it should not be denoted $\mathrm{arctanh}$. – J. M. isn't a mathematician May 29 '13 at 7:44

Write anything that you think is clear and understandable.

Amendment by Lord_Farin: When you feel your notation may be less than universally understood (given the tags on the question etc.) cq. ambiguous, it's always good to say e.g. "where (symbol) denotes (special function)".

• I hope you are okay with this amendment to your answer; it felt more natural than adding another one. You may reformat to your liking. – Lord_Farin May 21 '13 at 18:33