Orientation of 3 points $\mathbb{R}^{2}$ in logarithmic scale

Question

Is there a fast way (without evaluating $e^x$) to calculate orientation of 3 ordered points in $\mathbb{R}^{2}$ in logarithmic scale?

In other words we need to calculate sign of the following determinant: \begin{vmatrix} 1& e^{a_x}&e^{a_y} \\ 1& e^{b_x}&e^{b_y} \\ 1& e^{c_x}&e^{c_y} \end{vmatrix}