The right thing in this case is not to trigger this message at all.
When you run this in a shell
find /var/log/myfile.*.txt -type f
/var/log/myfile.*.txt is expanded by the shell; find gets expanded object(s) or literal /var/log/myfile.*.txt if there is no match. The latter case triggers No such file or directory.
You can create a dummy file first: touch '/var/log/myfile.dummy_name.txt' (compare elephant in Cairo). This "solution" is not really elegant though.
A better solution is to make find handle the pattern:
find /var/log/ -type f -name "myfile.*.txt"
where double-quotes prevent globbing in the shell (compare this). In this case * is handled by the find itself because -name supports such patterns.
The above approach however can match myfile.*.txt in any subdirectory of /var/log/ as well. If your find supports -maxdepth, use it:
find /var/log/ -maxdepth 1 -type f -name "myfile.*.txt"
If not, see this: Limit POSIX find to specific depth?
Your OS probably already uses logrotate(8) to manage logfiles.
In my Debian logrotate runs daily because of /etc/cron.daily/logrotate; it should be similar in Ubuntu. I can create a custom config in /etc/logrotate.d/ and manage arbitrary logfiles this way. Consider this approach.