I was reading the book Riemann's Zeta Function, by H. M. Edwards, page 42, where is a theorem that estimates the number of roots of the $\xi$ function
$$\xi(s)=\Gamma\Big(\frac{s}{2}+1\Big)(s-1)\pi^{-s/2}\zeta(s),$$
built with the Gamma function and Riemann's Zeta Function. The estimation is inside or on the circle $|s-\frac{1}{2}|=R$. Below is the screenshot of the theorem and the proof. 
There is one step of this proof that I don't understand. How is proven the last inequality? The one that establish that $$\frac{2}{\log 2}R\log R+ 2R-\frac{\log\xi(\frac{1}{2})}{\log 2}\leq 2R\log R.$$