What is the math behind this transformation on exponents that are logarithms?

exponentiation logarithms

By definition, $a=b^{\log_b(a)}$ and $n=b^{\log_b(n)}$. Therefore $$a^{\log_b(n)}=(b^{\log_b(a)})^{\log_b(n)}=b^{\log_b(a)\cdot\log_b(n)}=(b^{\log_b(n)})^{\log_b(a)}=n^{\log_b(a)}.$$

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