Begin with d=rt, where distance is in miles, rate is in miles per hour, and time is in hours. It took the woman 2 hours to paddel 2 miles up stream. Since the effect of her paddling on her overall net speed is constant and the river current rate is constant, we have the equation
Solving for r_w yields
Again, return to d=rt to model her downstream path. She paddled downstream for 2 miles this time making the stream current additive, but did so for an unspecified amount of time. The equation to represent this is
Substituting (2) for r_w yields
Now examine the log. Return to d=rt and model what happens. The log travels 1 mile, but for 1+t hours. 1+t comes from the fact that the log is floating downstream while the woman is still paddling upstream for 1 hour and then paddling downstream for the unspecified amount of time, t. The equation for this is
Equations (4) and (6) can now be used to solve for r_c. The work is below
Thus the rate of the current is 1/2 miles per hour, or at least I think so.