# Kelly's maximum for G(f)

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?

• Remember that $q=1-p$ – noob2 Jun 12 '17 at 14:04
• Perfect thanks, noob2. I had noted the relationship, but needed your nudge to connect the dots. – markm Jun 12 '17 at 18:06