This result is a special case of
R5358: Gibbs' inequality for simple relative entropy with
\begin{equation}
q_1 = q_2 = \cdots = q_N = \frac{1}{N}
\end{equation}
This is because in this case we have
\begin{equation}
\log_a q_n
= \log_a \frac{1}{N}
= \log_a 1 - \log_a N
= - \log_a N
\end{equation}
and therefore
\begin{equation}
- \sum_{n = 1}^N p_n \log_a q_n
= - \sum_{n = 1}^N p_n \log_a \frac{1}{N}
= \log_a N \sum_{n = 1}^N p_n
= \log_a N
\end{equation}
$\square$
