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

[download]

In reply to Re: RegEx Question - I want a line that contains this, but not this by rubasov
in thread RegEx Question - I want a line that contains this, but not this by mmontemuro

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.