Integral in Edwards' book

Question

I would like to understand the proof that $$\lim_{\tau\rightarrow \infty} \lim_{T\rightarrow \infty} \int_{a-iT}^{a+iT}\frac{\log(1-s/(\sigma+i\tau))}{s^2}x^s ds =0$$ where $a$ and $\sigma$ are real numbers with $a>\sigma$. This is a step from Edwards' book "The Riemann zeta function" on page 29 (1974 edition) in the proof of the principal term for the analytic formula for $\pi(x)$ that is the main result in Riemann's 1859 paper. The book just states "it is not difficult to prove..." and mentions the Lebesgue dominated convergence theorem, but I am having trouble putting a bound on the integrand because of the complex logarithm.