Proceeding directly from the definitions and applying results
we have
\begin{equation}
\begin{split}
H_a(X)
& = - \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a \mathbb{P}(X = x) \\
& = - \sum_{x \in \{ 0, 1 \}} \mathbb{P}(X = x) \log_a \mathbb{P}(X = x) \\
& = - \mathbb{P}(X = 0) \log_a \mathbb{P}(X = 0) - \mathbb{P}(X = 1) \log_a \mathbb{P}(X = 1) \\
& = - \frac{1}{2} \log_a \frac{1}{2} - \frac{1}{2} \log_a \frac{1}{2} \\
& = - \log_a \frac{1}{2} \\
& = - (\log_a 1 - \log_a 2) \\
& = - \log_a 1 + \log_a 2 \\
& = - 0 + \log_a 2 \\
& = \log_a 2
\end{split}
\end{equation}