I have a function $h(x, y) = g(r)$, with $x = r \cos \theta$ and $y = r \sin \theta$. I was able to find a formula for $\Delta h$:
$$\Delta h = \frac{\partial^2 g}{\partial r^2} + \frac{1}{r} \frac{\partial g}{\partial r} + \frac{1}{r^2} \frac{\partial^2 g}{\partial \theta^2}$$
I need to prove that the family of functions for which $\Delta h = 0$ is only $h = c \log (r) + d$ where $c, d$ are constants. We just started doing Calc 3 in this class so I don't really know what tools I can use from here.
Ummm, what you want to prove is not quite true. It is true for the special case of $\partial_\theta g=0$. But if $g$ is allowed to vary in $\theta$, nope. So I would first assume $\partial_\theta g=0$. Then demonstrate that your solution actually solves the "radial" part of the equation. Then, since that part is a second order linear equation, it can have only two independent solutions. You have two independent solutions ($c\log(r)$ and $d$), so they have to be the ones.