Back up Full Recovery Model Without Log Backups
When you take full database backup it will not take all the log file, and it seems you have some confusion about copyonly option.

If the issue you are facing is free desk space then you can refer to Is it possible to restore sql-server bak and shrink the log at the same time?
if your transaction log is growing too fast then you can start troubleshooting  and check this URLtoo

if the database is used for development environment then you can change it to simple mode this will truncate transaction log for you, but if you are on production site then do not do any changes to transaction log 

see what will happen if you truncate logs