Take x=1/(Wpi) where W is the number of microstates: \[\begin{align} \begin{aligned} \ln \left[\frac{1}{Wp_i}\right] &\le \frac{1}{Wp_i} - 1 \quad \textrm{for}\;\textrm{all}\; i = 1,\dots, ...Take x=1/(Wpi) where W is the number of microstates: ln[1Wpi]≤1Wpi−1foralli=1,…,W.W∑i=1piln[1Wpi]≤W∑i=1(1W−pi)−W∑i=1pilnW−W∑i=1pilnpi≤1−1=0−kbW∑i=1pilnpi≤kblnW. Moreover, the equality holds if and only if \(p_…