Complex logarithm on the negative real axis ($\log x=\log |x|+(2k+1)i\pi ,x<0$)

A continuous choice of complex logarithm on a negative real interval has a fixed integer $k$ in the displayed expression. The choice with argument $\pi$ is the upper-bank boundary value of the principal logarithm; the usual analytic principal-branch domain itself excludes the negative axis. Consequently, integrating a real weight times that logarithm adds $(2k+1)i\pi$ times the integral of the weight to the real logarithmic integral. The branch choice must be stated.