Logarithm definition

Question

Can I ask you why this equality holds ?

$3^{log_4{n}} = n^{log_4{3}}$

I was looking for a written explanation on how to get $n^{log_4{3}}$ given $3^{log_4{n}}$.

Thanks in advance.

Answer 1

$3^{\log_4n}=4^{\log_43\log_4n}=n^{\log_43}$

Answer 2

$$3^{\log_4n}=\left(n^{\log_n3}\right)^{\log_4n}=n^{\log_n3\log_4n}=n^{\frac{\log_43}{\log_4n}\cdot\log_4n}=n^{\log_43}$$

Answer 3

$y:= \log _4 (n)$, then $4^y=n.$

$3^y =(4^y)^{\log_4(3)};$

$3^y = 4^{(y \log_4(3))} = 4^{\log_4(3^y)} = 3^y.$