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It is clear for me how to proof the following rules:

$ a^x\cdot a^y=a^{x+y}$

$(a^x)^y=a^{xy}$

for $a\in \mathbb{R}$ and $x,y \in \mathbb{Z}$.

But i don't understand the step for concluding the same rules for $x,y \in \mathbb{Q}$.

I know from school, that these rules are applicable also for $x,y\in \mathbb{R}$ as well. How can one show this?

Averroes2
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  • Which definition of "power" are you using? There are several, and the proof of your statements for rational and reals will depend on the definition. Note that good definitions for rational then real exponents are not easy, and some (like defining $\ln(x)$ as an integral, $e^x$ as the inverse function, and $a^x$ as $e^{x\ln a}$) make no distinction between integral, rational, or real exponent. – Rory Daulton Nov 06 '16 at 10:25
  • $\mathbb{N}:= \cap $ all inductive sets. Based on this was $\mathbb{Z}$ and $\mathbb{Q}$ defined. – Averroes2 Nov 06 '16 at 11:26
  • That comment is too vague to base a proof or demonstration. Do you mean for $x\in\mathbb N$ that you define $a^0=1$ and $a^x=a\cdot a^{x-1}$, then $a^x=\frac{1}{a^{-x}}$ for negative integers, then $a^{p/q}=\sqrt[q]{a^p}$ for rationals? If so, how do you handle $x\in\mathbb R$? As I said, there are multiple definitions. – Rory Daulton Nov 06 '16 at 11:33

1 Answers1

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We can define the natural logarithm function for positive real numbers by $$\ln(x)=\int_1^x\frac{dt}{t}.$$

This function is one-to-one and has all real numbers in its range, so its inverse can be defined for all real numbers, we we call it $e^x$, or $\exp(x)$. From the properties of logarithms, we deduce that $\exp(x+y)=\exp(x)\exp(y)$. Once we have this function defined, we can define exponentiation as:

$$a^x=\exp(x\ln a).$$

From this definition, the exponent rules follow:

$$\begin{align} a^xa^y = \exp(x\ln a)\exp(y\ln a) &= \exp(x\ln a+y\ln a)\\ &=\exp((x+y)\ln a)\\ &=a^{x+y} \end{align}$$

$$\begin{align} (a^x)^y = \exp(y\ln a^x) &= \exp(y\ln(\exp(x\ln a)))\\ &=\exp(y(x\ln a))\\ &= \exp(xy\ln a)\\ &=a^{xy} \end{align}$$

Note that all of this applies to $a>0$ in $\Bbb R$. If $a$ is negative, these rules don't necessarily apply for all $x,y$.

G Tony Jacobs
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