I was just wondering why two numbers $a^x$ and $a^y$ are equal only if $x = y$ ? Which power-law is this?
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1Just bear in mind that $a=0,\pm 1$ break this rule... – abiessu Nov 21 '17 at 21:25
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1It's not a power law, it is injectivity of the function $f(x) = a^x$ (for $a$ positive and $\neq 1$). – Torsten Schoeneberg Nov 21 '17 at 21:25
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ah thank you very much!! – Msmat Nov 21 '17 at 21:33
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This is general property of injective function: $$f(x)=f(y)\Longrightarrow x=y$$ In this case your function is the exponent function $f(x)=a^x$, $(a\ne 1, a>0)$
nonuser
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Wow!! I didn't thought of that. Because i looked at $a^x$ only as a number. Now I see, thank you. – Msmat Nov 21 '17 at 21:29
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1If you think that this answer in useful for you than you can accept the answer by clicking on $\surd$. – nonuser Nov 22 '17 at 17:25
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It is true only if $a>0$ and $a\ne 1$ and , in this case, it is a consequence of the fact that the function $y:\mathbb{R}\to\mathbb{R} \quad y=a^x$ ,is one-to-one.
Emilio Novati
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