Finding explicit terms for $U(x,y)$ and $V(x,y)$

Question

We are given $u = U(x,y)$ and $v = V(x,y)$ and $x = e^u \cos v$ and $y = e^u \sin v$. We are to find explicit terms form $U(x,y)$ and $V(x,y)$. I have tried some bruteforce-techniques, but I am lagging behind in the course and am not sure how to proceed. You may assume familiarity with partial derivatives, gradients and most standard techniques of differentiation in your answer.

Answer 1

Formally: $$ \tan v = \frac{y}{x} $$ and therefore $$ u = \log \frac{x}{\cos v} = \ldots $$