Question

You are in a boat on a perfectly calm lake. A floating log passes near. You grab the log and put it in the boat. During this process, does the level of the lake rise, fall, or stay the same?

Answer 1

The water level should remain the same

Answer 2

any reason behind it?

Answer 3

Like how the polar ice can melt and not change sea level because the volume of water locked up as solid water ice is "already in" the ocean. I believe that this would translate to your system. The volume of both boat and log are "already in" the lake, even though the log might be high and dry at this point.