One approach, How can I run Log::Log4perl under mod_perl?, says synchronize access to an appender using Log::Log4perl::Appender::Synchronized, or use a different logfile for every process

Another approach might be to use a database appender like Log::Log4perl::Appender::DBI instead, and use pid/appname as part of the PatternLayout

Then you can schedule a cronjob to create dumps of database or ...