The portfolio log-return gap

Prove equation (2.2): for weights $w_i \ge 0$ summing to 1, $\ln\sum_i w_i e^{r_i} = \bar r + \frac12\sum_i w_i(r_i - \bar r)^2 + O(\max_i|r_i - \bar r|^3)$, where $\bar r = \sum_i w_i r_i$. Evaluate the exact gap and the approximation for two equally weighted stocks with log returns $-15\%$ and $+5\%$.