If I have a bunch of tar.gz files, and I know these all contain just log file data, and I just want to find and list particular lines that match a given regex pattern -- that is, I don't care which log file(s) are involved, I just want the matching lines -- I would either use a shell command like this:

gunzip -c *.tgz | grep target_regex > target.hits
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(You might need an extra option, "--binary-files=text" on the grep command, depending on which version of grep you have.)

Or else I'd write a perl script like this:

#!/usr/bin/perl

use strict;
use PerlIO:gzip;

for my $file (<*.tgz>) {
    open( T, "<:gzip", $file ) or do {
        warn "open failed on $file: $!\n";
        next;
    };
    while (<T>) {
        print if /target_regex/;
    }
}
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(And in the latter case, I'd probably use @ARGV to set the path for where to find the tgz files, what regex to apply, and/or whether to read the list of tgz file names from STDIN, so that the script serves a general range of uses.)