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 $n$?
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 $n$?
By definition, $a=b^{\log_b(a)}$ and $n=b^{\log_b(n)}$. Therefore $a^{\log_b(n)}=(b^{\log_b(a)})^{\log_b(n)}=b^{\log_b(a)\cdot\log_b(n)}=(b^{\log_b(n)})^{\log_b(a)}=n^{\log_b(a)}.$
Use change of base on $\log_b(n)$ and write it as $\frac{\log_a(n)}{\log_a(b)}$. Thus, $a^{\frac{\log_a(n)}{\log_a(b)}}=n^{\frac{1}{\log_a(b)}}=n^{\log_b(a)}$.