@fabian There isn't one, even historically, as far as I know (I am a mathematician, but not a historian of mathematics). The reason, I suppose, is historical happenstance. Most mathematical functions don't get their own symbol (sin, cos, tan, tanh, cosh...) and the log family is no exception; I think it is the square root that is the exception, whose first symbol was invented by Regiomontanus probably for his convenience.