Which of the following is false?
A) $100n \log{n}=\left(\frac{n \log{n}}{100}\right)$
B) $\sqrt{\log{n}}=O(\log \log{n})$
C) If $0<x<y$ then $n^x=O(n^y)$
D) $2^n \neq O(nk)$
What is the correct answer here? I'm sure $C$ and $D$ are right but not able to judge between $A$ and $B$. This is a GATE Exam question.
For B and similar problems with mulitple logs, exponentials, and powers, a trick I've found helpful is taking an extra log or two of both sides.
Instead of asking whether $\sqrt{\log n} = O(\log \log n)$, investigate whether $\frac{1}{2} \log \log n = O( \log \log \log n )$, which is of course false since the right side has more logs and thus will grow asymptotically slower. Thus, the original claim is also false.