Does 2 ^ O(log log n) = O(log n)?

Question

Does 2 ^ O(log log n) = O(log n)? Could you explain how to test this relationship?

I have tried substituting O(log(log n)) by C1 log(log n) and log n by C2 log n to find the relationship between them. When I graph the functions it seems that the statement is true, but I get stuck in the mathematical proof at one point and am not sure how to proceed.

Answer 1

No. Assuming a base-2 log, then

2^{log log N} = log N,

but 2^{10 log log N} is also in 2^{O(log log N)}, and

2^{10 log log N} = (2^{log log N})¹⁰ = (log N)¹⁰

...and that is obviously not in O(log N)