/etc/bashrc

export PROMPT_COMMAND='RETRN_VAL=$?;logger -p local6.debug "$(whoami) [$$]: $(history 1 | sed "s/^[ ]*[0-9]\+[ ]*//" ) [$RETRN_VAL]"'

or

function log2syslog
{
   declare COMMAND
   COMMAND=$(fc -ln -0)
   logger -p local1.notice -t bash -i -- "`who am i | awk '{print $1}'`:${USER}:${COMMAND}"
}
trap log2syslog DEBUG