Difference between complexity logn and log(sqrt(n))

Question

considering

log(sqrt(n)) = (1/2)log(n)

And if for asymptotic analysis we don't consider the constant terms so, is O(log(sqrt(n))) is as good as O(log(n))?

As per my understanding log(sqrt(n)) will grow slowly in comparison to log(n) if we increase the size of n. But I am not able understand the glitch in moving power of (1/2) at front? Is it just this that factor 1/2 only slows down the rate?

consider the case when we have log(n*n) represented as 2log(n) , and log(n)?

Answer 1

It is the same asymptotically:

O(log(sqrt(n))) = O(log(n^1/2)) = O(1/2 log(n)) = O(log(n))

Answer 2

You are right, O(log(sqrt(n))) is the same as O(log(n)) by the reasoning given in your question.