Fig.: to sleep very soundly

After a day of very hard work, at eight p.m. Max was already sleeping like a log

Dopo un giorno di duro lavoro, alle otto di sera Massimo dormiva già come un ghiro

As soon as I got home from my last, long and exhausting trip, I hit the bed right away and slept like a log for twelve hours non-stop

Al mio rientro dall'ultimo lunghissimo ed estenuante viaggio sono andato immediatamente a letto e ho dormito come un ghiro per dodici ore filate

It probably comes from early pioneer days in North America when lumber-jacking was popular. These burly men would cut down trees with basic tools, such as band saws. The trees were then cut into big logs. This was physically demanding work, so by the end of a day the lumberjacks were exhausted and would fall sound asleep, immobile like their heavy logs.