And there's just 1 logarithm, which has the property log(x^y) = y log(x). You don't need the ones with a different base.
The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is syntactic sugar. (If you define exp on C, even sin and cos...)
I guess a different notation could have some meaning pedagogically, math notation is incredibly inconsistent at times, but there really is no "deeper truth" here.
However in the cases where you need to be careful most of stuff you'd use the more general notation for wouldn't be applicable anyway, you'd have a high chance of writing down an expression that has no unique value, or can't even be evaluated.
Why do you think this? It doesn't seem to fit historically or formally. 3^2 is a more elementary object than anything built with exp, and there's often no natural notion of exp(A) of a function A even when finite powers like A^3 are defined. exp(A) is defined with a power series that may not converge.
I don't really conceptualize them as the same kind of object, to be honest. I'm aware exponentiation is more fundamental, but the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers.
When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge, for example, (-2)^2 is well defined and equal to (-2)(-2), but (-2)^(1/2) isn't, unless you relate it to the exponential function.
Since the OP was about a notation proposal for working with real numbers, in that specific context I believe the more natural interpretation is to relate everything to the complex exponential and work your way up from there.
But it works when "this" is an action that does not have a sensible notion of being applied a fractional number of times, or an infinite number of times, which is a very large and important set of actions indeed.
> When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge
Yes of course, but mere continuity of the inputs doesn't pick out exp from any other base you might choose besides e. You do not have to relate it to the exp function.
> Since the OP was about a notation propos...
My specific objection was to your suggestion that a^b is syntactic sugar, which suggests notational convenience that does not reflect what's going on "under the hood".
Yes, I agree that 'syntactic sugar' is not the word we are looking for here, I have no objections to your comment.
I disagree, I do think there's a deeper truth. It seems to me that you have already internalized and understood it very well; that is the goal.
would you at least agree that there's beauty here? and this notation does make it more apparent; or as you say "could have some meaning pedagogically"
If this can help students "see the light", then sure, but I'm not entirely sold on the idea that the notation is actually the hardest part.
This is similar (with a different twist) to the various ideas going around that we should stop using base 10 and think in hexadecimal. It might be better in some absolute sense, but it's not something it's worth spending any energy into.
I'm just not sure I see much from this, but obviously I'm not the "intended target", so please take this with a pinch of salt.
That's why math notation is used instead of the extremely wordy sentences that were used centuries ago.
It involves getting a geometric understanding of e to the pi i, and 3blue1brown explains it better than I could:
https://www.youtube.com/watch?v=F_0yfvm0UoU [6.28 min]
https://www.youtube.com/watch?v=v0YEaeIClKY [3.14 min]
The question is, can we find a notation that is much better at teaching the intuitive relationship between these 3 operations. Because notation isn't just about formalizing. Notation is about enabling better understanding. And for a large part, good notation should help teaching. Especially something like exponents, roots, and logarithms which are taught to many. People who happen to not get this probably stop doing math, and we need more people who can do math it seems.
Maybe I misunderstood your comment but it really sounds like you're saying that only the natural logarithm has this property, but in fact it's true with every base:
log(x^y; b) = log(x^y)/log(b) = y log(x) / log(b) = y log(x; b)
Yes I noticed their use of a comma (and "which" rather than "that"), in fact that's why I added the disclaimer at the start of my comment about possibly misunderstanding theirs. But sandwiching a mention of that property in the middle of making the same point twice only makes sense if it's a supporting argument.
That's not true. The natural logarithm is the only one whose derivative is 1/x but the identity you wrote is true with any base.
People had been using logarithmic tables to do multiplication centuries before Napier came up with "e".
Also to the best of my knowledge there's no logarithm tables that predate Napier (1550-1617), and when you try to calculate those tables you naturally stumble upon the natural logarithm (or at least a very close approximation), as the easiest way to create a logarithm table is to start with the powers of something like 1.00000001.
To add on that, we need only one logarithm in the sense that all other then follows. For any base b, we have
log(x; b) = log(x)/log(b)
log(x; b) = log(x; c)/log(b; c) = log(x; d)/log(b; d)
The second equality is why you can ignore c and d, and just pretend there's one logarithm when you do
log(x; b) = log(x)/log(b)
Also exponentiation is a way more fundamental property than the logarithm so it's weird to place its notation on the same footing.