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(When one says several things in a question, then several things may get answered and others neglected. Hence this posting overlaps with one of my earlier ones, but (I hope) this one will be short, simple, and narrowly focused.)

In what contexts in mathematics, at any level of sophistication, do products of logarithms, all to the same base, arise naturally?

(I know that I've come across them a few times while doing so routine calculus problems, but I can't remember anything specific about it.)

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If any other tags that I've overlooked would be suitable here (as I can't help but suspect) could others add those? –  Michael Hardy Oct 4 '12 at 22:04

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In computer science, the notion of polylogarithmic growth appears naturally when analyzing time/space complexity of algorithms.

There also exists the notion of quasi-polynomial time, which is a bit more rare though.

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I have no idea if this sort of thing counts $$\int_0^1 \left|\log_e x\right|^n\, dx = n!$$

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