Why does $n^{\ln {\ln n}} = ({\ln n})^{\ln n}$?

Question

Note: this is (part of my solution to) a homework question. Please DO NOT tell me the answer!

I am trying to compare the following functions:

$$n^{\ln {\ln n}} \qquad\qquad ({\ln n})^{\ln n}$$

It appears that they are equal (assuming $n > 1$), but I have absolutely no idea why this would be the case.

I am missing something really obvious and I have been hitting my head on this for about 45 minutes. A hint would be appreciated.

Answer 1

Try put the two expressions into the form $e^{(\cdots)}$ and compare.

Formula : $x^y = e^{y\ln(x)}$.

Answer 2

Take the logarithm of both expressions.

$\ln(x^y)=y\cdot\ln(x)\implies\ln(\ln(n))\cdot\ln(n)=\ln(n)\cdot\ln(\ln(n))$.

Answer 3

Use substitution \begin{align} n&=\exp(t). \end{align}