as others have said, syslog forwarding is the usual method.
However, if the router has a panic (chipset drivers, OOM etc) then it can cause the network stack to stop before it spits out the useful log lines. It should then reboot after 3s (I think).
If it is doing that and you want to get really deep into it, you'll need to use serial console/logging via the UART pins on the board. That's a whole different ball game!