Please be aware to only use this feature for gathering files of a reasonable size (I would say as a rule of thumb, less than 50MB in size) when using the Download button. This feature/page is accessible to users with Administration access only, but downloading files that are too big can cause performance issues for other users or even cause an out of memory error and force the app server to stop.
Many of the log files 'roll over' to keep the sizes down anyway, but some do not, so care needs to be taken. E.g. it's possible on a busy day for app-access logs to grow large, or with some other custom logging enabled such as the capturing of Verbose GC (garbage collection) logs, they could grow to hundreds of MB in size.