You're asking for more than a snippet there...

my $date = { set the date pattern you are looking for... }
my $in_date = 0;
while (my $line=<LOG>){
    chomp $line; 
    if ( ($in_date == 0 ) && ($line =~ m@SYNC@) && ($line =~ m@STARTED
+@){
           $line =~ s/\#//g;
           $line =~ s/^\s+//;
           $line =~ s/\s+$//;
           my @f=split(/\s+/,$line);
           shift @f;
           shift @f; 
           my $rdate = join(" ",@f);
           if ( $date eq $rdate ) {
              $in_date=1;
           }
     } elsif ( $in_date && ($line =~ m@SYNC@ ) && ( $line =~ m@FINISHE
+D@ ) ) { 
          $in_date = 0;
     } elsif ( $line =~ m@HEFFILE@ ) { 
           printf "%s\n",$line;
     }
}
[download]

That's the basic gyst of it. You may want to tweak it and I won't guarantee there aren't bugs...