In one of my textbooks, the expansion of a logarithm product is proved using integrals.
$$\ln xy = \ln x + \ln y\iff \int_1^\left(xy\right)dt/t$$
$$\ = \int_1^xdt/t + \int_x^\left(xy\right)dt/t$$
Then, let $ u = t/x $ and substitute in the second integral:
$$= \int_1^x dt/t + \int_1^y du/u = ln x + ln y $$
While the expansion is rather quaint, I find the u-substitution difficult to follow:
If $u = t/x$ then $du/dt = 1/x$ and $du = dt/x $ ... I am unsure how $ du/u $ is obtained for substitution.
Any help or hints are appreciated!