What does $2^x$ really mean when $x$ is not an integer? We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so please let me know what you think.

If this helps:
For all n:
$$2^n=e^{n\log 2}$$
This is a smooth function that is defined everywhere.
Another way to think about this (in a more straightforward manner then others described):
We know 
$$a^{b+c}=a^ba^c$$
Then say, for example, $b=c=1/2$.  Then we have:
$$a^{1}=a=a^{1/2}a^{1/2}$$
Thus $a^{1/2}=\sqrt{a}$ is a number that equals $a$ when multiplied by itself.
Now we can find the value of (for some p and q), $a^{p/q}$.  We know:
$(a^x)^y=a^{xy}$
thus 
$(a^{p/q})^{q/p}=a^1=a$
Other exponents may be derived similarly.
