Define F(x) = integral from 1 to x of dt/t. Show (from
first principles) that F(xy) = F(x) + F(y).
this is the most interesting bonus problem. we are proving the most
useful property of what will turn out to be the logarithm function.
if i can use Int(a,b) to mean the integral from a to b,
Int(1,xy)[dt/t] = Int(1,x)[dt/t] + Int(x,xy)[dt/t]
now do a miraculous substitution: t = xu (x is a constant here)
and dt = x du. notice what this does to the bounds in the integral:
Int(1,xy)[dt/t] = Int(1,x)[dt/t] + Int(1,y)[xdu/xu]
= Int(1,x)[dt/t] + Int(1,y)[du/u]
F(xy) = F(x) + F(y)