- #26

Mute

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Even with technicalities about proving a^0 = 1 versus defining it, your argument there is circular. You want to "prove" that [itex]a^0 = 1[/itex], and then proceed to do so by writing it as [itex]e^{x\ln a}[/itex], and then set x = 0 and say, "Oh, e^0 = 1, so a^0 = 1", but you didn't prove that "e^0 = 1"! You used the result you were trying to prove in order to prove it!One can define exponentials in terms of the family of standard functions and it falls naturally that anything to the power zero is one. Exponentials are defined as follows:

[tex] n^x = e^{x\ln{n}} [/tex]

Now put x = 0 and see what happens.