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What definition of $a^b$ operation is the most popular and standartized: Exponentation or Power? Is any difference between them?

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up vote 2 down vote accepted

First I'll interpret your question as: what, if any, is the difference between the formula $a^b = e^{b\ln(a)}$ (for positve $a$) and the operation of computing $a^b$ by multiplying $a$ by itself $b$ 'times'. I hope this is what you mean by the question, in which case here is an answer.

To start things off, there is no formal difference between the two in the sense that they will always yield the same result. Before going on, let's just make sure how to make sense of $a^b$ when $b$ is an arbitrary real number. One first defines $a^b$ for natural $b$ and then extends the definition for integer $b$, and then for rational $b$ by preservation of rules of powers that holds for natural $b$. For an arbitrary real $b$ one takes a sequence of rational numbers $q_n \to b$ that converges to $b$ and then define $a^b$ to be the limit of $a^{q_n}$. This requires some work to show independence of the chosen sequence of rationals.

Now, pedagogically, this definition is very appealing. It serves to show how definitions can be extended algebraically and analytically and it appeals to our sense of what it means to take powers. However, it's quite complicated.

On the other hand, defining $a^b = e^{b\ln(a)}$ is a very short definition (provided one knows, or believes to know, what $e^-$ is. However, it is not a very enlightening definition. As a formula it is very easy to work with which is why it is the one most often used.

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Actually, to even make $a^b = e^{b\text{ln}(a)}$ a non-circular definition, you'd first need to define $e^x$ and $\text{ln}(x)$ in a way that doesn't require powers. Since that obviously rules out power series, youd probably have to resort to $\text{ln}(x) = \int_1^x \frac{1}{y}dy$, and then define $e^x$ as the inverse function. This looks rather inconvenient... –  fgp Oct 15 '12 at 13:16
    
or one can define e^x as the solution to the DE y'=y and y(0)=1 and define ln as its inverse. Which of course is still inconvenient. –  Ittay Weiss Oct 16 '12 at 3:47
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It differs even visually, if you iterate. You can iterate in the way $a^b, (a^b)^b, ((a^b)^b)^b, \ldots$ or $ a^b, a^{a^b}, a^{a^{a^b}}, \ldots $.
Written in the form of a sequence, I'd write the first one as $a_0, a_1=a_0^b, a_2=a_1^b, \ldots ,a_{k+1}=a_k^b $ and the second one as $ b_0, b_1=a^{b_0}, b_2=a^{b_1}, \ldots, b_{k+1}=a^{b_k} $

To have the things also verbally distiguished, I'd say in the first version " a to the b 'th power", and in the second version " b exponentiated with basis a " .
And to make it even clearer, for the repeated application: " a (repeatedly) to the b 'th power", and in the second version " b (repeatedly) exponentiated with basis a ".

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