Usually, log messages are saved in dynamic memory. They will disappear once you reboot your system. You can have a copy of the log messages by setting up a Syslog server. carry out the step-by-step instructions provided on this page for remote log server Asus router.
The procedure given in this step will be suitable for the Asus RX3042 router:
The following steps are for the Asus RT-AC66U router: