# Simplify $n^{\log\log n / \log n}$

I am interested in solving logarithmic expressions but I cannot do this.

what does this expression simplify to?

$$n^{\log \log n/\log n}$$

Assuming $n \neq 1$, and let

$$y = n^{\log \log n/\log n}$$

$$\log y = \frac{\log \log n}{\log n} \log n = \log \log n$$

$$\Rightarrow y = \log n$$

(Spelling correction done)

Note that $n = e^{\log n}$, so $n^x = e^{x\log n}$.

Then $n^{\log\log n / \log n} = e^{(\log\log n / \log n)\cdot\log n} = e^{\log\log n}$.

$e^{\log \log n} = \log n$.