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I am doing a high order log transformation (e.g. $\log(\log{X+1}) = X_t$) but don't know how to express it concisely. Is there a convention I can use to express the order of a log transformation?

nmasanta
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Daniel Scott
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    $O(\log^2(x))$? – David P Jan 18 '20 at 00:29
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    @ David Peterson - is that correct? Maybe I'm being stupid but isn't the definition of $ \log^2(x) \ = (\log(x))^2$, not $ \log(\log(x))$ – Adam Rubinson Jan 18 '20 at 00:41
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    Analytic Number Theory has many nested logs, and they're usually written explicitly (like $\log(\log(x))$). So it is that there's even a joke around that makes fun of it: "What sound does a drowning analytic number theorist make? Log log log log..."! Well, in any case, you can just create a notation for it and leave it well-explained so there's no doubt about it's meaning. –  Jan 18 '20 at 00:42
  • Related discussion: https://math.stackexchange.com/questions/150546/what-does-log2x-mean – angryavian Jan 18 '20 at 00:43
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    @AdamRubinson $f^n(x)$ can mean either $f(f(\cdots (f(x)))$ or $(f(x))^n$. In a particular field, or by defining it, it should be clear which context is being used. – David P Jan 18 '20 at 02:44
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    consider using your own notation, by example you can set $$J_n:=\underbrace{\log\circ\log\circ\ldots \circ\log}_{n\text{ times }}$$ or something similar – Masacroso Jan 18 '20 at 03:32
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    In any case, I think it's better to be clear than concise. So just avoid using ambiguous notation. Write either $ \ (\log(x))^2\ $ or $ \ \log(\log(x))\ $. If you were to read or write $\log^2 (x),\ $ take it to mean $ \ (\log(x))^2\ $ by default, because that is by far it's most common meaning. – Adam Rubinson Jan 18 '20 at 12:39
  • No clue what the downvotes are indicating. – Daniel Scott Jan 18 '20 at 20:25

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I could be wrong, and it's been less than 24 hours since you posted your question, but the fact that no one in the community seems to know of such a mathematical symbol indicates that there probably isn't any common notation for what you're after.

In any case, making mathematical symbols go from 11 characters to 6 characters - or whatever - should be the least of your worries as a mathematician. Although, if you're into concise programming, then I can see why you asked your question- but I doubt most programming languages have a common symbol for what you're after.

I like to be as clear and unambiguous as possible, and I try not to sacrifice clarity for some fewer characters.

So just write $\log(\log(x))$, and if the order is of say, 30, you can write:

"$\log(\log(...\log(\log(x))))\ $ where there are $30 \ (nested)\ \log$s."

Adam Rubinson
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  • Thank you for a pragmatic answer. This indeed was related to a precise programming issue. I am often expressing order 10 or higher. – Daniel Scott Jan 18 '20 at 20:27
  • Have you tried also asking on stackoverflow: https://stackoverflow.com/ . That's where I would ask a programming-related question. – Adam Rubinson Jan 18 '20 at 21:19