johngg's previous answer works perfectly, I only would like to show you a more general technic to express "I want my regex to match this, but do not match that". The key is to use a positive and a negative look-ahead together:
#! /usr/bin/perl
use strict;
use warnings;
my $re_yummy = qr/^L[13]/;
my $re_yucky = qr/TEXT/;
while (<DATA>) {
print if / \A (?= .* $re_yummy ) (?! .* $re_yucky ) /x;
}
__DATA__
L1 04:10:07.915 LOG: Want this Line1
L3 04:10:07.915 LOG: Want this Line2
L1 04:10:08.024 LOG: TEXT. Do not want this Line3
L3 06:37:58.163 LOG: TEXT. Do not want this Line4
L3 07:02:36.921 LOG: Want this Line5
L4 08:02:30.910 LOG: Do not want this Line6
L5 08:02:36.943 LOG: Do not want this Line7
L6 09:02:38.811 LOG: Do not want this Line8