TITLE: What is the math behind this transformation on exponents that are logarithms?
QUESTION [10 upvotes]: I understand that 
$$a^{\log_b(n)} = n^{\log_b(a)}.$$
What is the math behind this transformation that allows you to swap the $a$ and $n$?

REPLY [11 votes]: By definition, $a=b^{\log_b(a)}$ and $n=b^{\log_b(n)}$. Therefore
$$a^{\log_b(n)}=(b^{\log_b(a)})^{\log_b(n)}=b^{\log_b(a)\cdot\log_b(n)}=(b^{\log_b(n)})^{\log_b(a)}=n^{\log_b(a)}.$$