The Balance Druid should have never been called that, it was supposed to be called ugly chicken thingy, but the devs read the wrong patch note stating it had been balanced before Vanilla release and accidentally called it Balanced with a typo instead.
It is two part word play of context of this game. One is the term we all know - a log of combat actions and results, but may also show the results of the activities of nearby characters. yes ?
But a “log” also is a subpart of a felled tree trunk.
And since druids have also a tree form. Which originally was available only in restoration spec (now is a flavour glyph also all other druid specs can use), so a druid in a treeform in melee is a log that fights - a … combat log!
In modern BfA context this can also modified to :
What do you call a kul tiran druid ? … because their forms are wooden. But the original jest is just word play around druid tree form and word that means two things.