# How to prove this theorem for exponential equations

Prove that if $$a^x=a^y a∈R$$ then $$x=y$$ as in the exponential equation $$2^x=2^3$$

How can I prove this theorem, if you know what I mean

• This is not always true, for example if $a=1$. – David Apr 8 at 2:09
• What if 1^x=1^4? it's true – Luis Gerardo Zárate Apr 8 at 2:11
• $x=4$ is not true if $x=5$. – David Apr 8 at 2:19
• It's also not necessarily true in the complex numbers. For example: $$e^{i\pi} = e^{i 3\pi} = ... = e^{i(2k+1)\pi} = -1$$ whenever $k$ is an integer. The key point in this discussion is you need to specify very, very, very clearly the framework, restrictions, and assumptions in which you are working. – Eevee Trainer Apr 8 at 2:24
• a∈R, I mean only for real numbers – Luis Gerardo Zárate Apr 8 at 2:28

$$a^x = a^y$$