What is the $(\lg n)$-th root of $n$? I am looking for the answer of the $(\lg n)$-th root of $n$, that is, $\sqrt[\lg n]{n}$. What is the answer and what log property should I use here? Please assume base as $2$ and $n$ as a natural number.

Take the $\log$ of this expression, to get
$$\log (n^{1/\log(n)}) = \frac{1}{\log(n)} \log(n) = 1.$$
This means that: 
$$n^{1/\log(n)}=2$$ 
or generally, the base of your $\log$.
