nat log to syslog server

The snat/nat function does not log anything, as this could produce a huge amount of data, however you capture and send this information by adding a simple irule to any virtual servers that you need to produce logs for.

You could use standard logging, and then set up logging (syslog-ng) to relay to a remote server, or you could use High Speed Logging (HSL) to send syslog messages directly from tmm to the receiving server:

You then just need to send a message containing whatever information you require, such as IP::local_addr, IP::remote_addr, TCP::local_port, TCP::remote_port.

For example:

when LB_SELECTED {
    log local0. "[IP::client_addr]:[TCP::client_port] -> [IP::local_addr]:[TCP::local_port]; [HTTP::host][HTTP::uri]"
}

Another way you could do it would be to use a request logging profile:

https://devcentral.f5.com/questions/configuring-a-request-logging-profile

Does that help ?