How to find definition for $a^r$ when r is any real number?

I was foraying into an advanced high school maths textbook and found this question -- For exponent $a^r$,

If $r = 0$ then $$a^r = 1$$ If $r$ is a natural number, then $$a^r=a^{r-1}.r$$ If $r$ is a negative integer, then $$a^r={\frac1{a^{-r}}}$$ If $r$ is a rational number, so that $r = p/g$ in lowest form, then $$a^r = a^{p/g}$$

The question asks to give a definition for all $a^r$ when $r$ is any real number. My problem with this one is that it's the first time I've encountered a question such as this and I'm not sure where to start.

Please point me in the right direction. I know some properties of real numbers (commutative, distributive etc.) but how do I use them here to make a definition of $a^r$ for all real $r$?

Solution 1:

As @lulu mentioned, it is difficult to answer, for example, I do not know what theory you are assuming true to ask your question. You can see it in three quite natural ways:

1. If you assume that the exponentials and logarithms are well defined with the properties that are already known, then it would be easy to see this:

Let $r\in \mathbb R$, then we define $a^r:=e^{r\ln a}$, then for example $$a^ra^s=e^{r\ln a}e^{s\ln a}=e^{(r+s)\ln a}=a^{r+s},\; etc.$$

2. If we assume that you have already carried out sequences, we can rely on the density of rational numbers:

Let $r\in \mathbb R$, then exists a sequence $(r_n)\subset\mathbb Q$ such that $r_n\to r$. SO we can define $a^r=a^{\lim_{n\to \infty}r_n}$, for example in this case let $r,s\in \mathbb R$, then exists $(r_n), (s_n)\subset\mathbb Q$ s.t. $r_n\to r$ and $s_n\to s$. Hence $$a^ra^s=a^{\lim r_n}a^{\lim s_n}=\lim a^{r_n}\lim a^{s_n}=\lim a^{r_n}a^{s_n}=\lim a^{r_n+s_n}=a^{\lim(r_n+s_n)}=a^{r+s},\;etc...$$

3. Assuming that you have done the definition of the real numbers in an axiomatic way and not completely from the rationals, that is, you already have the axiom of the supreme:

Let $r\in\mathbb R$, then we can define $a^r:=\sup\{a^q:q\in\mathbb Q\wedge q\leq r\}$ or $a^r:=\inf\{a^q:q\in\mathbb Q\wedge q\geq r\}$.

I hope this can help you.