Use system instead of exec. And you probably want to remove the newline from the serial:

#!/usr/bin/perl

use strict;
use warnings;

open(SERIALS, "serials.txt") or die("can't open serials.txt");
my @serials = <SERIALS>;
for my $serial (@serials) {
    chomp $serial;
    print "now processing $serial";
    system("echo grep $serial logs* > $serial.log");
}
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