How does $\ln(x)^{\ln(x)}$ behave? Can it be shown to be theta to any simpler, more familiar function? (polynomial, exponential, log-linear)?
Asked
Active
Viewed 73 times
3
-
The domain should be $(1,\infty)$, yes? – ajotatxe Aug 04 '19 at 09:33
-
yes, naturally otherwise it is undefined – Confused Soul Aug 04 '19 at 09:41
1 Answers
5
For $x>1$,
$$(\ln x)^{\ln x}=e^{\ln x\ln\ln x}=x^{\ln\ln x}$$
so it is greater than every polynomial, but lesser than every exponential.
ajotatxe
- 65,084
-
I was not sure if this would answer your question, thanks for the approval. – ajotatxe Aug 04 '19 at 09:59