# Expression of the fortune of a gambling problem

I was working on a gambling problem where the gambler with an initial fortune $$F_0$$ stakes a proportion of $$\theta$$ of his current fortune (Billingsley Ex. 7.5). The wager $$W_n=\theta F_{n-1}$$ and they ask to show $$F_n=\prod_{k=1}^{n}(1+\theta X_k)$$ and $$log F_n = n/2 \times [S_n/n \times log\{(1+\theta)/(1-\theta)\}+log(1-\theta^2)]$$. Here, $$S_n=X_1+\ldots+X_n$$.

I can prove $$F_n=\prod_{k=1}^{n}(1+\theta X_k)$$ by induction. But I cannot see how taking the log of this gives me the expression mentioned above. I have also tried it from backward without any luck. Any thoughts?