Log vs linear pots and the arduino
There are two associated issues here that underly your problem.

Firstly, with a log pot, you get a period of very small changes in resistance followed by a period of massive changes:



(* a "commercial" log pot is how log pots are usually manufactured - two linear traces of different resistances back to back).

So when turning you get "not much ... not much ... not much ... lots ... even more ... full".

Now couple that with how a motor works. A motor has a "stall current", which is higher than the "run current" (especially the "unloaded" run current).  This is the minimum amount of current required to overcome the inertia of the motor and start it spinning. This is, in effect, a lower threshold on the PWM value that you need to exceed before the motor will begin moving (once it's started moving you can go a little below that threshold and it will still move).

With a linear pot it's easy to get to that threshold and stay around it, increasing slowly above it, etc.  However, as you can imagine looking at the curve of the log pot above, if the "flatter" area of the pot is below the stall current threshold of the motor, the motor will do nothing until you are well into the sharper up-turn of the curve.  The effect of this is you will go from "stopped" to "quite fast" in one jump. It's very very hard to get even close to the stall current threshold.

For instance, if your motor needs a minimum of 30% duty cycle to start it running, that's about a 160 degree turn on the log pot above, compared to just 70 degrees of a linear pot, and the rest of the log pot's PWM range is compressed into a tight space.