Understanding how $() = C^{\log(x)}$ a polynomial function Is $f(x) = C^{\log(x)}$ a polynomial function?
I assume it must be because I learnt that you can rewrite this as $f(x) = x^{\log(C)}$ and given that $\log(C)$ is a constant, this looks like a polynomial function.  However, I'm struggling to understand intuitively why $C^{\log(x)}$ is polynomial, because it has a variable exponent in it, and I thought that all polynomial functions have a constant exponent in them.

That function is not a polynomial, but its asymptotic growth is polynomial, since (as you show) it can be written as a constant power of $x$. 
The fact that it can also be written some other way that does not look like polynomial growth is what makes the question interesting. 
The function
$$
f(x) = x^{2+ \sin(x)} 
$$
is not a polynomial but is $O(x^3)$.
