I have a quick question about simplifying these exponents and then comparing them:
$8^{\log_2 n}, 2^{3log_2(log_2n)}$ and $2^{(log_2(n))^2} $
I know the third one evaluates to $n^{log_2(n)}$, but I'm not sure how I would simplify the other two. I do know that $2^{log_2(n)}$ = n, but how could I use this to simplify the other ones because I don't think I'm simplifying them correctly. I also tried simplifying 8 into $2^3$, but I wasn't sure what to do from there.
Thanks!
Best Answer
$$8^{\log_2n}=2^{3\log_2n}=(2^{\log_2n})^3=n^3$$ $$2^{3\log_2\log_2n}=(2^{\log_2\log_2n})^3=(\log_2n)^3$$ Comparing these with $n^{\log_2n}$ we see that asymptotically $$2^{3\log_2\log_2n}<8^{\log_2n}<2^{(\log_2n)^2}$$