From the identity
\int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx,
deduce the value of \int_{0}^{\pi \slash 2} \log \sin x \, dx.
Putnamlogarithmstrigonometry