$log = "foo.log";
$pos_file = $log . ".pos";
$pos = 0;
if( open POS, "<$pos_file" ) {
  $pos = <POS>;
  close POS;
}
open FILE, $log;
seek FILE, $pos, 0;
...process file until eof...
$pos = tell FILE;
open POS, ">$pos_file";
print POS $pos;
close POS;