It's a silly song.

A log measures your speed through the water, but leeway is how the vessel moves relative to the wind.

If you are sailing North and the wind is from the West, then the vessel moves a little to the East under the press from the wind. Only a few degrees, usually, but it means that if you mean to get around a headland you can see in the distance which is just slightly to starboard you need to give it sea room or you could find yourself on a lee shore. You turn to port, so that your course is more to the West to give yourself a margin of safety. Even so with some vessels which have a shallow draught, such as the yachts for hire on the Norfolk Broads the amount of leeway they make is laughable.