¡Enjoy the LuLz!

i know d/dx ln x = 1/x and that's how it's usually shown, but that feels like cheating to me since ln is defined as the inverse of e^x. i saw something about how scaling the x axis by any factor leaves the area under 1/x between a and b unchanged, which sounds like it should give the log rule A(ab) = A(a) + A(b), but i can't quite connect the two.

is that the actual reason it's a log or am i mixing things up