First I'd use df to get the used space of a partition (du recursively walks the directory structure, which is rather expensive).

use strict;
use warnings;

sub partition_usage {
    my $partition = shift;
    my $line = (`/bin/df -B 1024 $partition`)[-1];
    return 1024 * (split(/\s+/, $line))[2];
}

# test it:
print partition_usage('/home/'), "\n";
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Then you should get a list of the log files you might want to delete. Then sort them by modification time. This example assumes that they are all in the current directory:

my @files = glob 'logfile_*.log.gz';
@files = reverse sort { -M $a <=> -M $b } @files.
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The the documentation of -M and sort for more details.

Then it's just a matter of walking through these files, and delete until you no longer exceed your size limit:

my $limit = 10 * 1024**3; # 10 GB
for my $filename (@files) {
    last if partition_usage('/var/log/') < $limit;
    unlink $filename or warn "Can't delete file '$filename': $!";
}
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