\begin{align} &\max_{p_1,p_2}\,\,\, \log_{2} \left(1+\frac{p_1 h_1}{\sigma}\right)+\log_{2} \left(1+\frac{p_2 h_2}{\sigma}\right)\\ &\text{s.t.}\,\,\, p_1+p_2\leq a \end{align}
where $h_1, h_2, \sigma$ and $a$ are positive rational constants and $p_1$ and $p_2$ are also positive rationals. I want to ask that how can we optimize $p_1$ and $p_2$ so that
$$\log_2 \left(1+\frac{p_1 h_1}{\sigma}\right)$$
and
$$\log_2 \left(1+\frac{p_2 h_2}{\sigma}\right)$$ are both integer values.