In Thorpe's paper, Thorpe derives the Kelly criterion
$$f^* = p - q$$
and plugs this into the equation
$$G(f^*) = p \times \log(1+f^*) + q \times \log(1-f^*)$$
to get the following expression
$$G(f^*) = p \times \log (p) + q \times \log (q) + \log(2).$$
I am struggling to derive this result. I am left with
$$p \times \log(p) + q \times \log(q) + p \times \log\left(1 + (1-q)/p\right) + q \times \log \left(1 + (1-p)/q\right).$$ I expect the last two terms somehow equal $\log(2)$ but I have been scratching my head for hours trying to get there. Can anyone show me how please?