It's been many years since I was first taught logarithms. So when looking at a proof that hinged on the fact that $a^{log_b n} = n^{log_b a}$, I thought that the proof was wrong. Actually, it was my fault: $a^{log_b n}$ is indeed equal to $n^{log_b a}$, I just forgot.

And so, I thought I would prove this fact. I am going to call it mathematical nonense, and its just short little lemmas that give me pleasure to prove and to reread.

\begin{aligned} a^{log_b n} &= X & \cr log_b (a^{log_b n}) &= log_b (X) \cr log_b n \cdot log_b a&= log_b(X) \cr log_b a \cdot log_b n&= log_b(X) \cr log_b n^{log_b a}&= log_b(X) & \cr n^{log_b a}&= X = a^{log_b n} & \cr \end{aligned}

Comments !