A little question about logarithm

Question

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.

score 6 · Accepted Answer · answered Mar 03 '13 at 15:18

6

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

answered Mar 03 '13 at 15:18

Brian M. Scott

score 4 · Answer 2 · edited Mar 03 '13 at 15:21

4

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

edited Mar 03 '13 at 15:21

Michael Hardy

answered Mar 03 '13 at 15:20

Ksenia

score 3 · Answer 3 · answered Mar 03 '13 at 15:21

3

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

answered Mar 03 '13 at 15:21

Michael Hardy

3 Answers3