$\log^2x$ = $(\log x)^2$ or $\log\log x$

Asked Apr 29 '20 at 09:23

Active Apr 29 '20 at 09:31

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When writing I found it a lot easier to say $\log^2x$ to mean $(\log x)^2$, but I didn't think about the ambiguity; because it is possible that can be interpreted as $\log\log x$.

My question is, what does $\log^2 x$ usually mean, or does it vary by source?

I'm trying to figure out whether it would be worth me getting in the habit of just avoiding the ambiguity completely.

edited Apr 29 '20 at 09:31

Jean-Claude Arbaut

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asked Apr 29 '20 at 09:23

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Of course, $\log^2x$ should mean $\log\log x$, but in practice it's usually $(\log x)^2$ by analogy with the dreadful $\sin^2x$. – Angina Seng Apr 29 '20 at 09:27
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I've always seen it used as shorthand for $(\log x)^2$, never for $\log\log x$ (sometimes in number theory $\log_2 x$ is used for the latter, to save on space when writing many repeated logarithms).
Nonetheless, I generally recommend getting into the habit of writing $(\log x)^2$ to avoid ambiguity.
– Thomas Bloom Apr 29 '20 at 09:27
No one writes that implying $\log \log(x)$. It’s always the square, no ambiguity there! (At least in analysis. Anyway, it’s better with the parentesis) – tommy1996q Apr 29 '20 at 09:27
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See also What does $\log^{2}{x}$ mean? and How to denote "powers" of a function?. – Jean-Claude Arbaut Apr 29 '20 at 09:30

$\log^2x$ = $(\log x)^2$ or $\log\log x$

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