Log-sum inequality

Theorem: For nonnegative numbers $a_1,a_2, \ldots, a_n$ and $b_1,b_2, \ldots, b_n$, \[ \sum_{i=1}^{n} a_i \log \frac{a_i}{b_i} \geq \left(\sum_{i=1}^n a_i \right) \log \frac{\sum_{i=1}^n a_i}{\sum_{i=1}^n b_i} \] with equality if and only if $\frac{a_i}{b_i} = \text{const}$.