Identity given in Shreve volume 1

Question

in a solution to a question about random walks (5.3 i), Part of the answer includes the identity:

$$\ln \frac{1+\sqrt{1-4 pq}}{2p}=\ln\frac{1-p}{p}$$

note that $p+q=1$ and $0<p<1/2<q<1$.

this does not seem true to me, even when I restrict the values of $p,q$ appropriately. What am I missing?

Answer 1

Using $q = 1-p$ we can work out the root as: $$\sqrt{1-4pq} = \sqrt{1-4p(1-p)} = \sqrt{1-4p+4p^2} = \sqrt{(1-2p)^2}$$

Taking the positive root reduces this to $(1-2p)$. This gives for the fraction:

$$\frac{1 + \sqrt{1-4pq}}{2p} = \frac{1 + (1-2p)}{2p} = \frac{1-p}{p}$$

This also holds inside the logarithm.