Does $a^b=a^c$ imply $b^a=c^a$?

Generally speaking $a^b = a^c$ implies that $b = c$ from which $b^a = c^a$ would then follow. But there are exceptions to this. If you can figure out what can go wrong then you can find out when $a^b = a^c$ but $b \ne c$ and then see if you then have $b^a \ne c^a$.

As a hint, examine the following proof: \begin{align*} a^b &= a^c \\ b\log(a) &= c\log(a) \\ b &= c. \end{align*}

This is a correct proof for most values of $a$ but something could go wrong for certain values of $a$. Can you spot it? Then for those certain values can you check if $a^b = a^c$ implies $b = c$ is still true?

Assuming $a,b,c>0$ and $a\ne 1$, $a^b=a^c$ is just $b=c$ ! (Taking the base $a$ logarithm.)

The converse is true, taking the $a^{th}$ root, $b^a=c^a\implies b=c.$

If $a=1$, $a^b=a^c$ is implicit.