What's the intuition behind non-integer exponents/powers Consider some $a \in \mathbb{R}$ and $x \in \mathbb{R}\backslash \mathbb{N}$. 
Is there some intuition to be had for the number $a^x$?
For example the intuition of $a^2$ is obvious; it's $a*a$ which I can think about with real world objects such as apples (when $a \in \mathbb{N}$). What about $a^{1.9}$?

Yet another intuition follows from the following formula: $$x^{\,y} = e^{\;y\log x}$$
(with appropriate restrictions of course).
