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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.

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    $\begingroup$ $\log_n$ seems fine to me... $\endgroup$
    – Jonny
    Mar 6, 2015 at 22:15
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    $\begingroup$ #4 denotes the fourth root of 1024, not the log of 1024 base 4. $\endgroup$
    – vadim123
    Mar 6, 2015 at 22:22
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    $\begingroup$ 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. $\endgroup$
    – Jonny
    Mar 6, 2015 at 22:24
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    $\begingroup$ This is a reasonable and interesting question and I'm surprised by all the downvotes! $\endgroup$
    – abnry
    Mar 6, 2015 at 22:42
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    $\begingroup$ 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. $\endgroup$
    – abnry
    Mar 6, 2015 at 22:44

4 Answers 4

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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.

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  • $\begingroup$ 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 ^,_ $\endgroup$
    – Dan W
    Mar 6, 2015 at 22:25
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    $\begingroup$ exponentiation does not have a unique inverse since it is a binary operation. $\endgroup$
    – vadim123
    Mar 6, 2015 at 22:27
  • $\begingroup$ Perhaps "inverse" was the wrong word. Regardless, it still completed the basic operator 'set' in my opinion. $\endgroup$
    – Dan W
    Mar 6, 2015 at 22:43
  • $\begingroup$ 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). $\endgroup$
    – Dan W
    Mar 6, 2015 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).

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  • $\begingroup$ 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. $\endgroup$
    – Dan W
    Mar 6, 2015 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

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    $\begingroup$ 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. $\endgroup$
    – Dan W
    Mar 7, 2015 at 1:16
  • $\begingroup$ @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). $\endgroup$
    – Vectornaut
    Mar 7, 2015 at 2:55
  • $\begingroup$ 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. $\endgroup$
    – Vectornaut
    Mar 7, 2015 at 2:56
  • $\begingroup$ 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! $\endgroup$
    – Vectornaut
    Mar 7, 2015 at 2:56
  • $\begingroup$ 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. $\endgroup$
    – Dan W
    Mar 7, 2015 at 3:13
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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.

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  • $\begingroup$ 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. $\endgroup$
    – Dan W
    Mar 6, 2015 at 23:00

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