I have a question. The question is to prove:

The weight $w \in A_\infty $if and only if $\frac{1}{|Q|}\int_Q w(x)dx \cdot \exp\left(\frac{1}{|Q|}\int_Q \log\frac{1}{w(x)}dx\right)\leq C$, for all cubes $Q\in \mathbb{R}^n$, where $C$ is a constant independent of $Q$ (and $w$?).

All other equivalent conditions of $A_\infty$ can be used. I really don't know how to get the point!