OK, we've had a talk about it and this is the best solution at the moment using the current state of the module:
use DateTime;
%OurLogs = get_the_time_from_the_log_file_as_a_hash();
$here = DateTime->new( time_zone => 'America/Indiana',
map { $_ => $OurLogs{$_} }
qw( year month day hour minute second ) );
$there = DateTime->new( time_zone => 'America/Los_Angeles',
map { $_ => $here->$_() }
qw( year month day hour minute second ) );
Then you can play around with however you want to format the $here and $there objects. Check the PODumentation for details.