It might be just a buffering problem. Are you sure you have waited long enough for the IO buffer (typically around 4k) to fill up?

Try flushing:

use IO::Handle;
...

  while (1) {
    print LOG "before sleep...";
    LOG->flush();    
    sleep(1);
    print LOG "after sleep...";
  }
[download]

(Also see Re: Troubles with do...while loop and sleep (line- vs. block-buffered) — or the classic "Suffering from Buffering")

(Update: moved LOG->flush() before the sleep — thanks ikegami)