If LogFile is not defined Zabbix (Agent and Server) will log it's messages to syslog. However, because the messages do not conform to RFC-5424 it is next to impossible to aggragate syslogs to a central server. Syslog-ng for example assumes the message comes from a host named "Zabbix" because the syslog messages are missing the real hostname and the string "Zabbix" from "Zabbix agent (daemon)[pid]" is used instead.