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Plus, minus, multiply, divide, and exponentiation all have symbols in math (+, -, *, /, ^ ) . But why isn't there the missing log symbol too? Here's how it would work:

4 ^ 5 = 1024 (as is standard for exponentiation)

1024 _ 4 = 5 ("_" is the new log operator!)

Look how much more elegant <1> is compared to <2>, <3> or <4>. We shouldn't need to do those 'hacks' to express the same thing:

1: 1024 _ 4 = 5

2: log(1024)/log(4) = 5

3: LogBase(1024,4) = 5

4: log4(1024) = 5

NB: It doesn't have to be an underscore symbol. It's just the first thing that sprang to mind.

Having a binary log operator would be useful for visually parsing the sum due to its conciseness. Additionally, using root symbols (for exponentiation's other inverse) eats up vertical space, and I think there's value in being able to express a sum on a single line. It's also easier to copy and paste a single line for use elsewhere when we use standard text symbols that are available on a keyboard.

Dan W
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    $\log_n$ seems fine to me... – Jonny Mar 06 '15 at 22:15
  • That's still using a name, rather than a one character symbol to express the same thing. – Dan W Mar 06 '15 at 22:17
  • It can (and in my opinion should) be treated as a symbol, just like + or -. There's nothing intrinsic to the laws of the universe to say it can't, right? I've edited my question to talk about the compactness of expressing math in single lines. – Dan W Mar 06 '15 at 22:21
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    #4 denotes the fourth root of 1024, not the log of 1024 base 4. – vadim123 Mar 06 '15 at 22:22
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    If you are so confident, then start using it and see if it catches on. I think the answer to your question lies more in the realm of the evolution of language than in Mathematics. – Jonny Mar 06 '15 at 22:24
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    The underscore is already used (for subscripts in LaTeX) and so are a surprisingly high number of other single characters on a keyboard. – Drone Scientist Mar 06 '15 at 22:24
  • @vadim12: Thanks, I removed that entry, and replaced it with Jonny's version. – Dan W Mar 06 '15 at 22:34
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    This is a reasonable and interesting question and I'm surprised by all the downvotes! – abnry Mar 06 '15 at 22:42
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    It also isn't off topic, even though it might be said it deals with the language of mathematics. Definitions and notation play a huge role in mathematics across the board and their choice impacts use. – abnry Mar 06 '15 at 22:44
  • @nayrb: Appreciated. I think historically, notation has evolved the way it has. But intelligent aliens (should they exist) may very well have a unique binary log symbol. – Dan W Mar 06 '15 at 22:44
  • They may, or someone may give a very good reason why it wouldn't turn out that way regardless. – abnry Mar 06 '15 at 22:45
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    @nayrb: I think the downvotes and closevotes are because the question looks more liker an attempt to convince the world to change notation than a genuine question looking for an answer. – hmakholm left over Monica Mar 06 '15 at 23:02
  • @HenningMakholm: I was looking for good reasons why the idea was flawed, as well as insight into sensible math notation generally, beyond its evolution as a series of somewhat arbitrary historical choices. If as a side effect, it turns out to be a good idea, and so takes a small step to change the way we use notation, that wouldn't necessarily be a bad thing anyway. – Dan W Mar 06 '15 at 23:07
  • Notation takes off when people use it and others find it convenient. It is as much a social construct as a mathematical necessity. Start using it. See whether others adopt it. Good mathematical notation clarifies ideas - my personal view is that this does not add additional clarity and is unlikely to be adopted. – Mark Bennet Mar 06 '15 at 23:57
  • @MarkBennet: I might just do that (and clarify what I mean whenever I do). It's not an unfair point, though I guess all new notation takes some getting used to. – Dan W Mar 06 '15 at 23:58
  • Related: "Alternative notation for exponents, logs and roots?". – Blue Aug 24 '16 at 14:10

4 Answers4

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^ is not really a mathematical symbol for exponentiation. The mathematical notation is to write the power as a superscript. ^ is just how the superscript is commonly (but not always) represented on computers in context where one has to stick to linear sequences of characters.

In any case, the answer is that there are many different functions that one might want a compact notation for, and only so many different symbols that one can reasonably expect people to remember. At some point using names becomes easier -- for example a name made up of letters can easily be looked up in an alphabetically arranged index; that is much harder if we just use some strange graphical symbol that doesn't have a conventional place in the alphabet.

Having a specialized notation for exponentiation is worthwhile because exponentiation is so common. In particular we use it to write down polynomials, which are very important functions.

Logarithms don't nearly reach the same level of importance among all the other functions one might also want specialized notations for.

hmakholm left over Monica
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  • Granted, something like + is more common than ^. But ^ is still incredibly popular, and if we have that, it makes sense to have its inverse function as a symbol too. That completes the set of six operators -,+ and /,* and ^,_ – Dan W Mar 06 '15 at 22:25
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    exponentiation does not have a unique inverse since it is a binary operation. – vadim123 Mar 06 '15 at 22:27
  • Perhaps "inverse" was the wrong word. Regardless, it still completed the basic operator 'set' in my opinion. – Dan W Mar 06 '15 at 22:43
  • Relevant: What's the inverse operation of exponents?. The page claims TWO inverses. At least though you can get one of the inverses with the existing ^ operator with the addition of the divide operator: 1024^(1/5). However, log is needed to obtain 5 from the other two numbers (1024 and 4). – Dan W Mar 06 '15 at 23:42
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All of the other examples (+,-,*,/,^) are binary operations.

Log is generally a unary operation. Either we are taking all of our logs base e, or base 10, or base 2. Only very rarely do we need various bases in a single passage. The proposed notation commits you to a binary notation, where the base of the log must be repeated every time. log(x) instead omits that base, once it has been specified. Hence the popularity of $\ln x$, $\log x$, $\lg x$ to denote these three bases respectively (although sometimes we use the middle one for something else).

vadim123
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  • You can still keep ln, log, or lg as separate common functions. You've already listed three common numbers used; having it as a binary operator is simply a generalization of that. At least in programming, I find I often have a need for an arbitrary base for the log operator. – Dan W Mar 06 '15 at 22:38
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But why isn't there the missing log symbol too?

The answer, as many people have pointed out, is "historical accident." The evolution of language is a rich, messy process whose details are hard to predict, control, or explain. For a glimpse of how our current mathematical vocabulary came to be, check out Florian Cajori's History of Mathematical Notations. Since Cajori died in 1930, his book is currently out of copyright in the U.S., so depending on where you live, you may be able to read and distribute it freely.

Notation for logarithms is discussed in the second volume of Cajori's book, paragraph 469. Intriguingly, Cajori doesn't document any notations like yours, with the logarithm appearing as a binary operator between the argument and the base. This may be because when people use logarithms, the base is often fixed, so it's inconvenient to mention the base at all.

A similar thing happens with exponents. In many situations (including exponentiation in Lie groups more complicated than the positive reals), it's easiest to work exclusively in (the generalization of) base $e$, and just write the exponential of $x$ as $\exp x$

I quite like your notation (although I'd strongly prefer a symbol other than _, since many people already use that for subscripts), and I hope it catches on! I'll even suggest a modification. If you change the order of the arguments so the base comes first, just like with powers, exponentiation and logarithms cancel neatly:

b ^ (b L x) = x

b L (b ^ x) = x

Vectornaut
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    Interesting suggestion to reverse the order. I was even thinking about using that suggestion for my calculator 'OpalCalc'. But I quickly tried + with - (and * with /), and it would seem having the base afterwards (like what I had) is consistent with that. Try it out, and see what you think: b + (b-x) != b - (b+x). Maybe there's some merit however to reversing the order for those too! So 5-4 would = -1 (not 1) in that system. – Dan W Mar 07 '15 at 01:16
  • @DanW: Ah, that's a good point! If you think of the inverse functions $\exp_b$ and $\log_b$ as analogous to the inverse functions $\operatorname{plus}_b$ and $\operatorname{minus}_b$, it does seem more natural to put the base on the right, so $\exp_b x$ would be written as ${}^xb$. Believe it or not, there's also a deeper reason you might want to write exponentials this way—one so compelling that I've seen at least one professional mathematician do it (though she is a bit of a joker). – Vectornaut Mar 07 '15 at 02:55
  • When we work with numbers, we're often using them as abstract descriptions of concrete sets of things: we say "five" instead of "five bananas." Correspondingly, many arithmetic operations can be seen as abstract versions of operations on sets. – Vectornaut Mar 07 '15 at 02:56
  • In particular, if the numbers $b$ and $x$ describe the sets $B$ and $X$, then the number $b^x$ describes the set of functions from $X$ to $B$, which is often written $B^X$. But since English is read from left to right, it feels weird to read $B^X$ as "functions from $X$ to $B$." The notation ${}^X B$ flows much more nicely! – Vectornaut Mar 07 '15 at 02:56
  • Be interesting to see his thoughts on 2-3 = 1, and 2/3 = 1.5 then to see if those too have any advantages over the usual interpretation. – Dan W Mar 07 '15 at 03:13
  • @DanW: Addition and multiplication, like exponentiation, can be seen very nicely as abstractions of operations on sets (disjoint union and Cartesian product, respectively), but I don't know of any simple set-theoretic interpretations of subtraction, division, or logarithms. – Vectornaut Mar 07 '15 at 04:14
  • That's part of why the abstraction from sets to numbers is useful: it lets us create inverses for addition, multiplication, and exponentiation. The price we pay for using these inverses is having to deal with weird kinds of numbers—negative, rational, and even real numbers—that are hard to interpret as sets. I've heard some beautiful interpretations of negatives and fractions in terms of things sort of like sets, but they're rather complicated. – Vectornaut Mar 07 '15 at 04:15
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As Henning Makholm notes, ^ is much more a "computer notation" rather than mathematical notation. And so is _. If you want to be strict about a "duality" between the mathematical notation of exponentiation as superscript, for logarithm you'd have to use subscript (which actually coincides with _ in LaTeX). So the inverse of $e^x$ (i.e. the natural logarihtm) would be $e_x$ in this "perfectly dual" notation. The obvious trouble with using subscript for logarithm is that there are great many other places/cases where subscripts are useful. So there would be a lot of confusion. Occasionally, but less often, superscripts too are used for denoting things other than exponentiation.

the gods from engineering
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  • We can be flexible with regards to what the symbol should look like - the underscore symbol was just the first thing that I thought of. For computer use, I'd rather use the symbols than superscripts or subscripts anyway. – Dan W Mar 06 '15 at 23:00