How to compare logs

Question

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!

Answer 1

$$8^{\log_2n}=2^{3\log_2n}=(2^{\log_2n})^3=n^3$$ $$2^{3\log_2\log_2n}=(2^{\log_2\log_2n})^3=(\log_2n)^3$$ Comparing these with $n^{\log_2n}$ we see that asymptotically $$2^{3\log_2\log_2n}<8^{\log_2n}<2^{(\log_2n)^2}$$

Answer 2

The only formula you have to know is that, $\forall a > 0$, $x \in \mathbb{R}$, $$a^x = e^{x \ln(a)}$$

Therefore, you have $$8^{\log_2(n)}=e^{\log_2(n) \ln(8)} = e^{3 \ln(2) \frac{\ln(n)}{\ln(2)}} = e^{3 \ln(n)} = n^3$$

Similarly, you have $$2^{3\log_2(\log_2(n))} = e^{3\log_2(\log_2(n)) \ln(2)} = e^{3\ln(\log_2(n))} = (\log_2(n))^3$$

And finally $$2^{(\log_2(n))^2} = e^{(\log_2(n))^2 \ln(2)} = e^{\log_2(n) \ln(n)} = n^{\log_2(n)}$$