Geometry of planar log-fronts

The log-front of two curves $P$ and $Q$ in a toric surface is the set of torus elements $\tau$ such that $\tau\cdot Q$ is tangent to $P$. Log-fronts generalize dual curves, wave fronts, and arise naturally in the theory of random surfaces. Our goal in this paper is to prove analogs of Plücker and Klein formulas for log-fronts.