TITLE: Is $n^{\log c} = c^{\log n}$ true?
QUESTION [5 upvotes]: Is $n^{\log c}$ the same as $c^{\log n}$? If so, please explain.

REPLY [7 votes]: $$
n^{\log_b c} = \Big(b^{\log_b n}\Big)^{\log_b c} = b^{(\log_b n)(\log_b c)} = \Big(b^{\log_b c}\Big)^{\log_b n} = c^{\log_b n}.
$$