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enter image description here

I have included this picture in its original form from my textbook here.

I think this is wrong because it contradicts the definition of $a^x$.

Because we define

$$a^x=e^{x\ln a}$$

where $a>0,x\in\mathbb R$.

Am I making a mistake or is the textbook wrong?

nonuser
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  • $a < 0$ in your example – Virtuoz Dec 18 '21 at 07:58
  • The expression is not even defined with your definition. Maybe the book uses a different definition? The question is borderline duplicate of What are the Laws of Rational Exponents?. – dxiv Dec 18 '21 at 07:59
  • @dxiv I think that the definition must be global and solid, right? – nonuser Dec 18 '21 at 08:02
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    @I'mastudent Solid, yes. Global/universal, not necessarily. Each book can tell the story in its own words, as long as the definitions are clearly stated and the language is consistent throughout. – dxiv Dec 18 '21 at 08:08
  • @dxiv but $a^x=e^{x\ln a}$ is not my definition.. – nonuser Dec 18 '21 at 08:17
  • @I'mastudent But you still aren't saying whether that definition comes from the same textbook that you are asking about. If yes, then the quoted line does not even make sense because the leftmost expression is not defined according to that definition. – dxiv Dec 18 '21 at 08:23
  • @dxiv I took the definition $a^x=e^{x\ln a}$ from wikipedia, not from the book.. – nonuser Dec 18 '21 at 08:26
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    @I'mastudent Find the definition in that book and edit that one into the question. – dxiv Dec 18 '21 at 08:28
  • @dxiv there is no definition for negative base in the textbook. Only example. – nonuser Dec 18 '21 at 08:34
  • @I'mastudent Then find a better textbook. You won't learn good math from a book that throws some random manipulation without defining what the starting point is even supposed to mean. – dxiv Dec 18 '21 at 08:44
  • @dxiv I understand, then can you suggest a definition such that the above is correct? – nonuser Dec 18 '21 at 08:52
  • @I'mastudent No sensible/self-consistent definition exists, see the linked q&a. – dxiv Dec 18 '21 at 09:06
  • @dxiv are you talking about global definition or do you say that there is no definition, such that the above is correct? – nonuser Dec 18 '21 at 09:25
  • @I'mastudent Guess you could come up with some contrived definition under which the posted statement holds true. However, such an arbitrary definition would not be 1) sensible, 2) self-consistent, and 3) "global", whatever that means. All that said, this is now drifting far away from the question as originally asked. I suggest you re-read the linked duplicate carefully, then - if you still have a question after that - ask a new question. Make sure to point out specifically how it's different from others asked before. – dxiv Dec 19 '21 at 07:52

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