Do we understand correctly that you currently have Adaxes store its logs in an external SQL database and will be moving to another one? If that is correct, you need to manually move the logs. Adaxes will not do anything about them and there is no such possibility. Using dedicated MS SQL tools should work just fine.
If you have Adaxes store its logs in the local SQLite database, switching to an external SQL database will have all the logs synchronized to it. As a result, you can decrease the retention period for log records in the local database: https://www.adaxes.com/help/CustomizeLogsStoragePeriod.