Log simplification

Question

Question Image

Now this is the question of asymptotic analysis in Algorithms; My question is $H(n)= n^\frac{1}{\log n}$, First i am taking log of this expression then can I apply power rule here and it will become $\frac{1}{\log n} \log n$ and Here we will have 1 only as answer.

Answer 1

Your image has $\log_2 n$ rather than $\log n$. If you take $\log_2$ of both sides you have

$$\log_2 H(n) = \frac{1}{\log_2 n}\log_2 n = 1.$$

But that means $H(n) = 2^1 = 2$. So if you're concluding that $H(n)$ is constant, you're correct.