This post is with the speed being constant relative to the bank.
A previous post (the one by daleks) stated that the stream would be going at half a mile per hour. But this must be incorrect, for the woman went 1 mile, met the log, and then went for another hour upstream. By now the log would have been half the way back to the start. The woman would then have to go back downstream for an hour to get to the 1 mile mark (for she would take 1 hour to go that same distance). When she reaches the 1 mile mark for the second time, she would have taken two hours to get back there and if the log was travelling at half a mile an hour, the log would have reached the start point but she would still be at the 1 mile mark, so this answer must be incorrect.
She travels 1 mile and meets a log.
From now the log travels 1 mile to the start.
She now travels at speed v for one hour upstream, travelling v miles.
She turns around and travels at speed v for one hour downstream, travelling v miles downstream.
She is now back at the 1 mile mark.
She now takes 1/v hours to get back to the start.
Therefore, both of them (the log and her) travel for 2 + 1/v hours before they get to the starting point.
Let the speed of the water be w.
speed = distance
It can be seen from this that if the speed is taken as constant relative to the bank, the speed of the river is linked to the speed of the woman.
If she was travelling at 1 mile per hour, the trip would take 3 hours. Hence, the stream must be travelling at 1/3 miles per hour.
If she was travelling at 2 miles per hour, the trip would take 2 and 1/2 hours. Hence, the stream must be travelling at 2/5 miles per hour.
If she was travelling at 10 miles per hour, the trip would take 2 1/10 hours. Hence, the stream must be travelling at 10/21 miles per hour.
The boundaries of the speed of the stream are:
upper 1/2 a mile per hour (as v tends
lower 0 miles per hour (as v tends to 0)
Of course, neither of these speeds can actually be reached, as they are the limits
Also, the speed of the stream must always be slower than that of the woman.