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In many U.S. middle schools and high schools, $\ln$ and $\log$ are treated differently, with the intent that $\log$ is equivalent to $\log_{10}$. However, in undergraduate courses and in the academic world, $\log$ always means $\log_{e}$, and $\ln$ is rarely, if ever used.

Is there a compelling reason to teach younger students the "notation convention" of $\log = \log_{10}$? Or should logarithms be taught always with a base so that students don't fall into this notation trap when they grow up?

It seems to be that there is no need to teach the notation $\ln$, and we should just teach $\log_{e}$ as $\log$ and use $\log_{10}$ if we need base-ten logarithm.

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    $\begingroup$ You are wrong about undergraduate courses. log is used for log base 10. Also, almost every calculator I've ever seen uses the log button to mean log base 10. $\endgroup$ – Sue VanHattum Dec 16 '19 at 19:50
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    $\begingroup$ In college, it is not true that $\log = \log_e.$ . Rarely is it used that way. $\ln$ is by far more pervasive in undergrad and grad students, meaning $\log_e$. In fact, any undergraduate student overlapping computer science and math will suggest that $\log$ should be interpreted to mean $\log_2$. It is possible that in a few nations, the preference at the undergraduate level deviate from this, but if that's the case, you should understand that doing so is an exception, not the rule. $\endgroup$ – amWhy Dec 16 '19 at 19:52
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    $\begingroup$ Hm, perhaps my empirical data is wrong... in all the undergraduate math/statistics/physics classes that my peers and I have taken, $\log$ has always been used to imply $\log_e$. $\endgroup$ – Nahmid Dec 16 '19 at 19:57
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    $\begingroup$ We already have two answers along this vein, but for my entire undergraduate career as an an engineer, ln was base e, log was base 10. For basically my entire engineering career, log is base 10. Only recently have I started dabbling in worlds where log means anything else. $\endgroup$ – Cort Ammon Dec 16 '19 at 22:46
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    $\begingroup$ The extensive discussion this question has provoked shows that it is clearly a good question about pedagogical practice, whose answer is not clear nor universally agreed. $\endgroup$ – Dan Fox Dec 17 '19 at 12:10
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Unrelated to US, in Germany the notation through school and university was quite consistent:

  • $\ln$ using base $e$
  • $\log$ using base $10$
  • $\log_x$ using base $x$

I may not know enough about the US education system, but if the definitions are not clearly homogeneous throughout the country, I would teach all three at school plus add a note that in some places the notations might differ.

In my opinion, telling students that a certain notation is set in stone, when it really isn't, might lead to more confusion later.

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    $\begingroup$ It's the same in the US. Outside of pure mathematical contexts or tools like Matlab, log(100) = 2. It's also a useful distinguisher for engineers, since most representations of power or frequency use base-10, whereas the actual time-domain signal analysis uses base-e. $\endgroup$ – MooseBoys Dec 17 '19 at 7:58
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    $\begingroup$ I went to school in Germany too, and we didn't use $\log$ for $\log_{10}$. We either wrote it out explicitly, or used $\operatorname{lg}$ for that. Then at university, we generally used $\ln$, hardly ever anything else. When we used $\log_2$ we wrote that out explicitly again. $\log_{10}$ was never needed. $\endgroup$ – leftaroundabout Dec 17 '19 at 10:46
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    $\begingroup$ @MooseBoys: Type $\log(100)$ on Matlab and you get $4.6052$. In Python, math.log$(100)$ returns $4.605170185988092$ (specifying a different base requires an optional second argument). In C the same. So outside pure math contexts the use of log to mean natural logarithm is quite common. $\endgroup$ – Dan Fox Dec 17 '19 at 12:04
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    $\begingroup$ @DanFox yes, though the historical reason for that is probably: Fortran had LOG=ln because maths, then C followed Fortran, then everybody followed C. So it does come from “pure maths” (actually rather from applied maths, physics simulations etc.), but it does make sense as I said in my answer. $\endgroup$ – leftaroundabout Dec 17 '19 at 13:23
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    $\begingroup$ @leftaroundabout dunno which school and university you attended but our math teachers at least gave during the introduction to logarithms the explanation that you usually write the base next to log - and then showing the 2 "exceptions" for their prominent and wide spread use .. lg / ln . AND NEVER I have seen the shortcut of log = logarithmus naturalis .. which is in fact ln That programming languages follow their own principles is well known. At least you can easily transform logarithms. $\endgroup$ – eagle275 Dec 17 '19 at 14:27
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You are wrong about undergraduate courses always treating $\log$ as $\ln$. To my memory, all of my undergrad chemistry and physics (not just general, but majors texts), engineering, calculus, diffyQs, and engine maths books use $\ln$\ for natural log and log for base 10. Add onto that everything I've seen professionally in oil exploration and the military. I have about 100 books like that on my shelves. Just picked ten of them at random and none of the ten used $\log$\ for Napierian logarithm.

Note also Sue's point on calculator buttons. This is a huge, common, commercial usage pattern.

Yes, I have occasionally seen the use of "log" to mean natural log, usually being able to detect it by something being "off" in the explication. I reviewed a recent paper from two writers from the Fed, who were using $\log$\ for natural log (without clarification) and recommended that they note their usage at the beginning. (I didn't need them to change their preferred notation, but did need them to clarify what they were doing.)

Personally, I find most equations to be based on natural logs anyhow. So in that case, use of $\ln$\ avoids any ambiguity. If you are planning to use $\log$ for base $10$ logs, it is probably also best to clarify your usage initially. Or if you insist on using $\log$ to mean natural log, clarify that on first usage. (But I would avoid, as less frequent.)

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I would argue that we should never use $\log$ for $\log_{10}$ anymore, only warn that this was historically often done. Sticking to the ISO convention is probably safest: $$\begin{aligned} \log_{e} \equiv \ln \\ \log_2 \equiv \operatorname{lb} \\ \log_{10} \equiv \lg \end{aligned}$$ I personally would use of the shorthands only $\ln$, and write out all other logarithms explicitly.

Further I would remark that most programming languages use log to mean $\ln$. This does make some sense, because

  • In a programming context you have lots more variable names so standard functions don't stand out the way they do in maths formulas; in particular something two-letter like ln would be not as clear to read.
  • The natural logarithm is by far the most important in serious applications. Computer scientists may disagree (in discrete maths, base-2 tends to turn up more), but then, they mostly write logarithms in big-O notation and there the basis doesn't actually matter because it is equivalent to a constant factor.
    The reason base-10 used to be common is that it makes calculation by hand easier, especially with a slide rule. But this is completely obsolete; today the base-10 logarithm is essentially irrelevant.
  • exp is generally understood to mean $x\mapsto e^x$, so using log for $\log_e$ gives a nice symmetry.

Still, in a maths context I'd continue to write $\ln$ for the natural logarithm, and only ever use $\log$ with an explicit base annotation.

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    $\begingroup$ Has "lb" ever been used outside the standards document? It seems like a standard that exists mainly to acknowledge that some people use base-2 logarithms. $\endgroup$ – chepner Dec 17 '19 at 13:14
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    $\begingroup$ "in particular something two-letter like ln would be not as clear to read" - What about \$ function in jQuery? It is a single-letter name. A name is just a name, it just has to be unique. $\endgroup$ – Rusty Core Dec 17 '19 at 17:29
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    $\begingroup$ I downvoted because I disagree. Calculators use log as a button, and it means log base 10. I think it's useful to keep that notation for as long as calculators do. $\endgroup$ – Sue VanHattum Dec 17 '19 at 19:03
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    $\begingroup$ @SueVanHattum then we should encourage calculator manufacturers to stop doing that. That doesn't happen by further perpetuating this stupid convention. $\endgroup$ – leftaroundabout Dec 17 '19 at 20:30
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    $\begingroup$ @leftaroundabout Because some people like having a dedicated device for calculations or for when they don't have access to a computer or smartphone with the appropriate applications installed. Additionally, doing calculations and graphing programmatically can require more in-depth work than using a dedicated device for that purpose. $\endgroup$ – JAB Dec 18 '19 at 1:14
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In mathematics, $\log$ means natural logarithm. So, as you become a mathematician, sometime during that process you must learn this.

That standards document is for natural sciences, not for mathematics. I guess mathematicians are too independent-minded for such things.

This would be seen, for example, in complex analysis and in real analysis.


Note: Of course we do not claim $\log$ is always used for natural logarithm (except perhaps among mathematicians). But neither can it be claimed that $\log$ is never used for natural logarithm.


Example:
W. Rudin, Principles of Mathematical Analysis ... 3rd edition, page 181.


Another example: the C++ draft standard

template complex log(const complex& x);
Notes: the branch cuts are along the negative real axis.
Returns: the complex natural (base e) logarithm of x, in the range of a strip mathematically unbounded along the real axis and in the interval [-i times pi, i times pi ] along the imaginary axis. When x is a nega- tive real number, imag(log(x)) is pi.
template complex log10(const complex& x);
Notes:
the branch cuts are along the negative real axis. Returns: the complex common (base 10)logarithm of x, defined as log(x)/log(10).

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    $\begingroup$ @SueVanHattum Most (but probably not all) working researchers in real or complex analysis will use $\log$ for $\log_e$, and rarely write $\ln$. I would guess this convention is introduced in graduate level real and complex analysis courses. $\endgroup$ – Steven Gubkin Dec 16 '19 at 23:31
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    $\begingroup$ -1 for lack of perspective. Different mathematical fields and textbooks use $\log$ in different ways. $\endgroup$ – Daniel R. Collins Dec 17 '19 at 5:18
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    $\begingroup$ I think the point is more that mathematicians don't tend to use anything other than base $e$, so use $\log$ and $\ln$ interchangeably. But other subject do use base $10$. $\endgroup$ – Jessica B Dec 17 '19 at 6:25
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    $\begingroup$ In professional mathematical usage, $\log$ essentially always means the natural logarithm. More precisely, it means the restriction to the real numbers of a branch of the complex logarithm. It's not just a question of choices of base. What occurs naturally is the entire function $e^{z} = \sum_{i = 0}^{\infty}\tfrac{z^{i}}{i!}$ and the problem of inverting this function. On the other hand, in engineering or economics, $\log$ often means the base $10$ (real) logarithm. $\endgroup$ – Dan Fox Dec 17 '19 at 12:00
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    $\begingroup$ However, that professional mathematicians use $\log$ to mean the natural logarithm does not entail that one should do the same when teaching students who are not future mathematicians, e.g. engineers. For such students, I find it causes fewer difficulties for students to write $\ln$. It costs me little, and requires less reeducation in a context where not much is to be gained by such reeducation. $\endgroup$ – Dan Fox Dec 17 '19 at 12:08
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My perspective on this (as a British computer scientist) is slightly different to the others already mentioned: my default expectation is that $\ln$ is the natural logarithm, $\lg$ is the binary logarithm, and $\log$ is used when the base is unimportant. Since I usually see logarithms inside a Landau $O$, the irrelevancy of the base is the most common case.

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    $\begingroup$ I have seen “lg” being used for base 10, “ln” for the natural one and “log” for a general one. $\endgroup$ – Martin Ueding Dec 17 '19 at 8:49
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    $\begingroup$ Knuth used lg for binary logarithm in The Art of Computer Programming. So there's always going to be some tension between ISO and seminal textbook as far as canonical notation. In the end, mathematicians will define the specific notation they will use in the context of their work, and mathematical readers would be well-advised to check for those notations before jumping to conclusions. $\endgroup$ – Matthew Daly Dec 17 '19 at 14:20
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    $\begingroup$ "$\log$ is used when the base is unimportant.": That's a really good point worth stressing. Because I think a lot of people make the mistake of assuming that there must be a specific base for the notation to be meaningful, causing them to miss the fact that logs are often base-agnostic. Indeterminate bases would be a great point to teach students early on, as the concept itself seems broadening. $\endgroup$ – Nat Dec 19 '19 at 17:58
  • $\begingroup$ @Nat: Interesting point. If one writes something like log(x)/log(2.0), any base for log will be essentially as good as any other provided both calls to log use the same one. I suspect the use of log as base 10 has to do with the fact that the easiest way to compute the natural logarithm was to compute a base-ten logarithm and multiply that by the the reciprocal of the base-ten log of e. $\endgroup$ – supercat Dec 19 '19 at 22:39
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This question is more difficult to answer than it appears. Part of the difficulty is that usage varies from area to area, from country to country, and from teacher to teacher. Part of the problem is that the answer might depend on the audience being taught. The summary of what follows is that: a. when teaching one should be up front about and make one's conventions clear; b. when teaching anyone other than future mathematicians one should either use $\ln$ or be very clear that one is using $\log$ to mean $\log_{e}$ (very clear means repeating endlessly); c. that the issue in professional practice is one of context - in mathematics there is not much need for other bases, while in contexts where the reference scale is important and it is natural to use some integer base such as $2$ or $10$, it is natural and convenient to write $\log$ to mean the logarithm in this base; d. in purely mathematical contexts $\log$ means the natural logarithm.

First, what is the logarithm? Among professional mathematicians, $\log$ means an inverse of the exponential function $e^{z} = \sum_{k \geq 0}\tfrac{z^{k}}{k!}$. More precisely, one calls any solution of $e^{z} = w$ a logarithm of $w$, and one writes $\log w$ for any such number. If $z = x + iy$, then $w$ solves the equations $e^{x} = |w|$ and $e^{iy} = w/|w|$. The equation $e^{x} = |w|$ has a unique solution given by the real logarithm, $\log|w|$. In general, $\log w = \log |w| + i \arg w$ where the definition of $\arg$ involves the choice of a branch cut in the complex plane. Here I have just paraphrased the definitions from the book of Ahlfors, which must be a reference as standard as any.

From a conceptual standpoint, over the real numbers, what are naturally defined are the exponential, as the aforementioned power series or as the solution of the initial value problem $\dot{x} = x$, $x(0) = 1$, and the real logarithm, as the integral $\log x = \int_{1}^{x}\tfrac{1}{t}\,dt$. These functions are inverses, and proving that is somehow the heart of a mathematically oriented calculus class. Which is the better starting point is perhaps a matter of preference. Personally, I like to start with the definition of the real logarithm as an integral, but on this point I change my mind every time I teach calculus.

If one goes through these proofs, one sees that working with another base is not terribly natural as it amounts to sticking some ad hoc constant factor in the initial value problem or integral. From this point of view, one has to do some work to define $a^{b}$ (or equivalently, $\log_{a}$), and this work amounts to the standard functional relations for either the exponential or the logarithm (the integral definition of the logarithm makes these straightforward). From a mathematical viewpoint there is no need to introduce any other notation than $\log$, nor is there often need to work with logarithms in other bases. Consequently, the mathematician often has difficulty seeing any need for introducing such things at all, and resists using notations such as $\ln$, and in some cases cannot understand why engineers and students and other lower forms of life like economists insist on writing such barbarities all the time.

The change in the base of the logarithm is natural when one thinks about scales, measurement, and the like, concepts where are nonmathematical or extramathematical in the sense that there is no logical necessity for them (one can always write $\log(b)/\log(a)$). Moreover, since $e$ is irrational, it is not a good choice operationally as a scale. Numbers such as $2$ and $10$ are better, more useful, more operational choices (are there any others?). In a context where $\log_{2}$ or $\log_{10}$ is used repeatedly, and no other choice is used, it is natural and convenient to drop the subscript and write simply $\log$ to mean the logarithm in the given base, and this is precisely what has happened quite generally. In such a context it sometimes becomes necessary to compute, and when doing so, the natural logarithm somehow creeps back because everyone writes $e^{x}$, and something like $\ln$ becomes standard to distinguish the natural logarithm (or does the notation abbreviate Naperian logarithm?) from the other in common use.

When teaching small children, it is impossible to speak of initial value problems and integrals, much less branch cuts and arguments, and there is no reasonable way to define $e$ or $\log_{e}$. On the other hand, ever small child knows $3^{2} = 9$ and $10^{2} = 100$, so it makes sense to define things like $\log_{2}9 = 3$ and $\log_{10}(100) = 2$, and more generally, logarithms with integer bases of integers make sense and are easy to define, and this is the natural starting point for working with logarithms when dealing with children. In such a context, it is entirely unnatural to write $\log$ to mean $\log_{e}$, since on the one hand one is not mentioning $e$, and on the other hand one cannot make any sense of it in any case. Worse, in high school, one tries to make sense of $e$ via some limiting procedure that usually starts from understanding exponentials/logarithms with rational bases. So high school teachers find the insistence of professional mathematicians that there is one true logarithm, $\log$, to be a bit difficult, and write $\log$ to mean $\log_{10}$, and $\ln$ to mean $\log_{e}$, when they get around to trying to give some meaning to the latter expression (if they ever do that at all).

Of course there are countries where $\log$ is written for $\log_{e}$ all along, and there are probably other countries where something else is written entirely.

And then along comes some signal processing engineer who wants to write $\lg$ for $\log_{2}$ because it is all she uses, and $\log$ already means $\log_{10}$ ... As a professional (hopefully) mathematician, this unnecessary barbarous new notation makes my blood boil, but latex already has the command \lg defined so it must be in common use whether I know it or not ... And electrical engineers outnumber mathematicians, so I am fighting a losing battle - and for what gain?

Now the student who has learned that $\log$ means $\log_{10}$ and $\ln$ means $\log_{e}$ gets to the university and the math professor keeps writing $\log$ to mean $\ln$ and it gets very confusing. The professor has three approaches:

  1. Write $\log$ to mean $\log_{e}$ and not worry about the matter further.
  2. Observe that mathematicians use $\log$ to mean $\log_{e}$, acknowledge that this can be confusing at first, and use $\log$ to mean $\log_{e}$.
  3. Observe that mathematicians use $\log$ to mean $\log_{e}$, but write $\ln$ because it causes less confusion and the engineering students in the room are never going to study complex analysis or Fourier series anyway (some of them should, but that's a different matter).

1 Is quite common, but not a good idea. In the past I always did 2, and think it is a perfectly defensible position. In recent years I have tended towards $3$, simply because it costs me nothing and I think costs the student nothing either (I am not teaching mathematics students, rather engineers). If I use $\log$ to mean $\log_{e}$, which I sometimes do because for me this usage is reflexive, I always mention that it means the base $e$ log not the base $10$ log. Students assimilate this rapidly. Were I teaching complex numbers, of course I would use $\log$ to mean the natural logarithm; in that context there's no other practice that makes much sense. Were I teaching in a context (physics, engineering, etc.) where the values of numbers had interpretations that mattered, then I would always specify the base, and would use $\ln$ when referring to the natural base.

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