Go back to the script that generates the log in the first place and store the epoch instead of the data you have. (or in addition to this stuff if you like to read it)
Then you can just subtract one epoch from the other to get total number of seconds. (The epoch is returned from time() )

So, for example, if your log now reads:

964644652       Wed Jul 26 13:50:52 2000
964644691       Wed Jul 26 13:51:31 2000
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You can parse this like so:

use strict; # Always.
my $logname = "log.txt";
open LOG, $logname or die "Failed to open $logname, $!";
my( $start, $stop );
while( $start = <LOG> )
{
  ($start) = split /\s/, $start;
  $stop  = <LOG> or die "No End Time in LOG!\n";
  ($stop) = split /\s/, $stop;
  my $time_in_seconds = $stop - $start;
  # Do something with $time_in_seconds
  print "$time_in_seconds\n";
}
close LOG or die "Failed to close $logname, $!";
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To get the log format I gave you I used:

use strict;
my $t = time(); 
print $t, "\t", scalar localtime($t)";
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Update: Tye Pointed out that split will take care of the remainder for me, so I took that out.

Also, you might want to note in the log which lines are 'start' and which are 'stop' so that you can catch errors (like failing to record the 'stop' time). Right now, you would only notice such an error if there is an odd number of occurances. (And only at the end of the script when it can't find an end time for the last entry.) Of course, then you would need to parse that extra bit of info from the log. But entries like:

START 964644691       Wed Jul 26 13:51:31 2000
STOP  964644691       Wed Jul 26 13:51:31 2000
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should not be hard to parse. just call split /\s/ as before, and make the recieving array ($tag, $epoch) or whatever.