Which one grows faster asymptotically: $\log(\log^an)$ or $\log^a(\log n)$

Question

Could someone explain to me which function grows faster?

$f(n)=\log(\log^an)$ or $g(n)=\log^a(\log n)$

Does $\log^a x$ mean $(\log x)^a$ or "take the log of $x$, $a$ times"? What did you try? Where did you get stuck? Did you look at our reference question? — David Richerby, Feb 14 '17 at 23:52

score 8 · Accepted Answer · edited Feb 15 '17 at 08:08

8

Assuming $\log^a n$ means $\log n$, raised to the power $a$:

Let $t = \log \log n$, then the first expression equals $t·a$, while the second expression equals $t^a$. For $a > 1$ the latter grows faster, for $a < 1$ the former is faster, and for $a = 1$ they are identical.

edited Feb 15 '17 at 08:08

Yuval Filmus

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answered Feb 15 '17 at 01:13

gnasher729

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Which one grows faster asymptotically: $\log(\log^an)$ or $\log^a(\log n)$

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