To view more than the Recent log messages at
admin/reports/dblog you need to use the error log or/and the access log of the webserver.
I'm not sure if in your case is Apache, NGINX or if you are using a webhosting service, some of them have an option to check this logs.
In the future, if you want to keep more messages at
admin/reports/dblog, you can adjust that at
admin/config/development/logging and increase the amount on the "Database log messages to keep" option. Careful with this option, a value like "1000000" could have a negative impact on the website performance.
If you don't have a shared hosting environment and you have ssh access to the webserver, I think the best option is to enable the syslog module. This module logs events by sending messages to the logging facility of your web server's operating system (syslog).