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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.

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up vote 6 down vote accepted

Yes. Let $x=\log_4n^2=2\log_4n$; then $2^x=2^{2\log_4n}=\left(2^2\right)^{\log_4n}=4^{\log_4n}=n$.

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Thanks, i will accept that in 10 minutes – bigO Mar 3 '13 at 15:19
@bigO: You’re welcome. – Brian M. Scott Mar 3 '13 at 15:24

You can also prove this using the fact that $\log_a b =\dfrac{\log_c b}{\log_c a}$.

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And more generally, $$ \log_b a = \log_{b^k} (a^k). $$

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