Which grows faster $n^{\log(\log n)}$ or $(\log n)^{\log n}$

Question

I usually just try playing around with properties of log, but it doesn't help in this case.

I tried to edit this to improve the format in the title, but I'm afraid I may have accidentally changed the meaning. Please check. — saulspatz, Mar 12 '21 at 00:53

score 2 · Answer 1 · answered Mar 12 '21 at 00:39

2

they are the same...............really...

answered Mar 12 '21 at 00:39

Will Jagy

139,541

@Alzebrian Not just similar -- the same. – Andreas Blass Mar 12 '21 at 00:52

score 2 · Answer 2 · answered Mar 12 '21 at 00:51

2

Take the log of each and apply $\log(x^y) = y \log x$ to obtain $$\log \left(n^{\log(\log n)}\right) = \log(\log n) \log n$$ and $$\log \left((\log n)^{\log n}\right) = \log n \log(\log n).$$ Because $\log$ is monotonic, the original expressions are also equal to each other.

answered Mar 12 '21 at 00:51

RobPratt

45,619

I don't understand step 2. It's is essentially saying log(log (n^2)) = 2 log(log n). Is that correct ? – Alzebrian Mar 12 '21 at 01:14
No, but $\log((\log n)^2)=2\log(\log n)$. – RobPratt Mar 12 '21 at 04:14

Which grows faster $n^{\log(\log n)}$ or $(\log n)^{\log n}$

2 Answers2