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From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 339). W. H. Freeman.

An exponential function is a function of the form $f (x) = b^x$, where $b > 0$ and $b \neq 1$.

Why is $b$ limited to the set of real numbers greater than zero?

goblin GONE
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StudentsTea
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    Well, what is $(-1)^{1/2}$? In $\mathbb{R}$, there is no sensible choice; in $\mathbb{C}$, there are two, namely $i$ and $-i$. In both cases, $x \rightarrow (-1)^x$ is not a well-defined function. – goblin GONE Aug 17 '14 at 07:25
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    Did you mean $b \neq 1$? – JimmyK4542 Aug 17 '14 at 07:27
  • @goblin has a good point. The beauty of complex analysis (and other math courses that follows calculus) is you get to see the reasons behind the madness. – Austin D Aug 17 '14 at 07:41
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    @Nomad any course that defers explaining the madness to a later course is not doing a good job. There is no madness in mathematics and there is no reason to present things is if there is madness with the disclaimer that things will be clarified later on. – Ittay Weiss Aug 17 '14 at 08:17
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    @IttayWeiss My point was not that mathematics is madness, but that is can seem like that to a student who hasn't taken some of the higher-level courses. I have several examples in my old Calc book where the author simply said, "The proof of this theorem is outside the scope of this course." It was one of those, "Take it on faith until we get to that section" things. – Austin D Aug 17 '14 at 08:26
  • Isn't it a bit draconian to exclude such a large swatch of numbers simply because they don't behave well with exponents between zero and one? Wouldn't we get a lot more bang for our buck if we let b < 0 back in if and only if | x | >= 1 or x = 0 ?

    Also--I can't wait to sink my teeth into Complex Analysis.

     – StudentsTea Aug 17 '14 at 08:31

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