#! /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