How does $\ln(x)$ behave when raised to itself?

Question

How does $\ln(x)^{\ln(x)}$ behave? Can it be shown to be theta to any simpler, more familiar function? (polynomial, exponential, log-linear)?

Answer 1

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.