Defined concisely: A king log is a ruler who is so peaceful and quiet in his reign that his subjects never feel his power.
It comes from a story attributed to Aesop about a group of frogs who became discontent with their life in the pond, and started begging Zeus to give them a king to rule over them. Zeus found their request humorous, and he sent them down a huge log which dropped into the pond.
At first they feared the log, but before long, since it never moved, they became less afraid of it and even began climbing on it. They cried out to Zeus again, begging him for a better king that would actually move. So Zeus sent them down a stork that promptly began devouring the frogs.
They pleaded with Zeus to take away the King Stork, but Zeus refused, saying that they had received what they requested, and should live with the consequences.
The moral of the story can be taken in different ways. Some would say, "Leave well enough alone." Others might say, "Be careful what you wish for."
There is a bit of humor that has been widely circulated, probably inspired by this story, of a stork attempting to eat a frog that is resisting him by choking the stork. It usually appears with the caption, "Never give up."
So depending on what you want out of your leaders, to have a "King Log" could be a good thing or a bad thing. It's nice to have a ruler who isn't afraid to step up and take action, but when the action comes at great cost to the citizens, perhaps having an inactive ruler might be a preferable option. It's also a good caveat about letting any one part of a government have too much power.