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

Question

Does $a^b=a^c$ imply $b^a=c^a$? I can't prove it mathematically. Can anyone show me how to prove this mathematically (if it's true), or disprove it?

(Note: I will delete if duplicate but I couldn't find any)

My attempt:

$a^b=a^c$

$a^{c-b}=1$

That means that either $a=1$ or $c-b=0$

$c-b=0 \implies b=c$

If $a=1$ then $c-b=0 \implies b=c$ is not necessarily true so $a=1$ is a counterexample for $b \ne c$.

Is this right? Is there another way to do it? Thanks in advance.

Answer 1

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?

Answer 2

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.