Let $X \in \text{Random}(\mathcal{X})$ be a D5723: Simple random variable.
Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.

Then

(1)	\begin{equation} - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a \mathbb{P}(X = x) \leq \log_a |\mathcal{X}| \end{equation}
(2)	\begin{equation} - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a \mathbb{P}(X = x) = \log_a |\mathcal{X}| \quad \iff \quad \forall \, x \in \mathcal{X} : \mathbb{P}(X = x) = \frac{1}{|\mathcal{X}|} \end{equation}