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:
- Write $\log$ to mean $\log_{e}$ and not worry about the matter further.
- Observe that mathematicians use $\log$ to mean $\log_{e}$, acknowledge that this can be confusing at first, and use $\log$ to mean $\log_{e}$.
- 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.
