Evaluate: ∫₀^(π/2) logtan x dx

Let $I=\int_0^{\pi/2}\log\tan x\,dx$. Using $\int_0^a f(x)dx=\int_0^a f(a-x)dx$: $I=\int_0^{\pi/2}\log\tan\left(\frac{\pi}{2}-x\right)dx=\int_0^{\pi/2}\log\cot x\,dx$ Adding: $2I=\int_0^{\pi/2}\log(\tan x\cot x)\,dx=\int_0^{\pi/2}\log 1\,dx=0$ Hence $I=0$.