Where is system.log in TimeMachine backup
Everything in /private is excluded by rule - including /var/log/system.log

tmutil isexcluded /private/var/log/*


More details on this are at On OS X, what files are excluded by rule from a Time Machine backup?

You would need another tool to back these up as I’ve not had luck using the tmutil removeexclusion /var/log/system.log command which might have worked if the exclusion was based on a file pattern and not the rule.