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

Question

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

what does this expression simplify to?

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

Answer 1

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)

Answer 2

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

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

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

Answer 3

$$\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$$ $$\frac{\ln\ln n}{\ln n}=\log_n(\ln n)$$ $$n^{\log_n(\ln n)}=\ln n$$