How to show that ${(\ln n)}^{\ln n}=n^{\ln(\ln n)}$

Question

How to show that $${(\ln n)}^{\ln n}=n^{\ln(\ln n)}$$

Attempt:

$y={(\ln n)}^{\ln n}$ then $\ln y=\ln n\ln(\ln n)$ what to do next?

Answer 1

HINT

We have

$${(\ln n)}^{\ln n}=e^{\ln n\cdot \ln (\ln n)}=(e^{\ln n})^{\ln (\ln n)}$$

and recall that by definition $e^{\ln n}=n$.

Answer 2

Hint : Take log on both sides

$${(\ln n)}^{\ln n}=n^{\ln(\ln n)}$$ Taking log gives $$ \ln\left({(\ln n)}^{\ln n}\right) =\ln\left(n^{\ln(\ln n)}\right)$$ $$ \ln({n})( \ln(\ln(n))= {\ln(\ln n)} \ln(n)$$

Answer 3

Alternatively: $${(\color{red}{\ln n})}^{\color{blue}{\ln n}}=(\color{red}{e^{\ln (\ln n)}})^{\color{blue}{\ln n}}=(e^{\color{blue}{\ln n}})^{\color{red}{\ln (\ln n)}}=n^{\ln(\ln n)}$$