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I have read the chapter up and down but I do not see how, I would like to not take anything from the book but start on e fresh example as I think that would help me to realise what is going on.

Im struggling to see how

$e^x$ is defined for all real x to define the arbitrary exponential

$a^x$ (where a>0) for all real x

Could someone show the connection

This is what is stated:

I have been shown this in the book: If r is rational $\text{Ln}\left(a^r\right)=r \text{Ln}(a)$, therefore: $a^r=e^{\text{rLn}(a)}$, I understand the first part($\text{Ln}\left(a^r\right)=r \text{Ln}(a)$), but I do not understand:

therefore $a^r=e^{\text{rLn}(a)}$

ALEXANDER
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1 Answers1

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$a^x=e^{x(\ln a)}$. This is how $a^x$ is defined in general.

voldemort
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  • I have been shown this in the book: If r is rational $\text{Ln}\left(a^r\right)=r \text{Ln}(a)$, therefore: $a^r=e^{\text{rLn}(a)}$, I understand the first part($\text{Ln}\left(a^r\right)=r \text{Ln}(a)$), but I do not understand therefore ..... – ALEXANDER Aug 23 '14 at 14:37
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    If $\ln x= y$ then $x= e^y$- this is by the definition of $\ln$. – voldemort Aug 23 '14 at 14:40