# A little question about logarithm

Can we say that $\log _4 (n^2)=\log _2(n)$? If that is the case, then $\displaystyle 2^{\log _4 (n^2)}=n$? Thanks.

Yes. Let $x=\log_4n^2=2\log_4n$; then $2^x=2^{2\log_4n}=\left(2^2\right)^{\log_4n}=4^{\log_4n}=n$.
You can also prove this using the fact that $\log_a b =\dfrac{\log_c b}{\log_c a}$.
And more generally, $$\log_b a = \log_{b^k} (a^k).$$