Easy question about logarithms

Question

Why is it true that $a^{\log_{b}n} = n^{\log_{b}a}$

Answer 1

$a^{\log_b n} = a^{\frac{\ln n}{\ln b}} = e^{\frac{\ln n}{\ln b}\ln a} = n^{\frac{\ln a}{\ln b}} = n^{\log_b a}$ for $a,b,n>0$.

Answer 2

By definition

$$x={\log_{b}n} \iff b^x=n$$

$$y={\log_{b}a} \iff b^y=a$$

therefore

$$a^{\log_{b}n} =a^x=b^{xy}=n^y= n^{\log_{b}a}$$