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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

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    $\begingroup$ This is not always true, for example if $a=1$. $\endgroup$ – David Apr 8 at 2:09
  • $\begingroup$ What if 1^x=1^4? it's true $\endgroup$ – Luis Gerardo Zárate Apr 8 at 2:11
  • $\begingroup$ $x=4$ is not true if $x=5$. $\endgroup$ – David Apr 8 at 2:19
  • $\begingroup$ 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. $\endgroup$ – Eevee Trainer Apr 8 at 2:24
  • $\begingroup$ a∈R, I mean only for real numbers $\endgroup$ – Luis Gerardo Zárate Apr 8 at 2:28
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$a^x = a^y$
x.log a = y.log a

log a = 0 iff a = 1.
Draw your own conclusions.

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