You have a couple of options:
1.) Increase the number of log lines stored in memory up from the default of 100. Upon a reboot however this will wipe out the log completely
2.) Store some log options to the flash disk. You usually only want to do this for really critical messages as writing to the flash drive takes up space, granted not a lot, and uses up CPU cycles. This will also impact performance on the router some as well if you are writing a lot to the disk.
3.) Download and install a syslog server somewhere and point all of your logs on the Mikrotiks to that. There are several options out there, some free and some pay for, we use Splunk. They have a free version that works well, but you won't want it to be on a direct public IP for the free version because they do not have any user accounts so anyone would be able to log into your server and see what logs you are getting.
Most syslog servers will index information for you so you can search through your logs for specific things.