Login to the CG-ONE and navigate to the message logs page for the function you wish to archive (eg. API Access)
Select the checkbox in the header row and a prompt will appear where you can 'Select all messages'.
Proceed to 'Select all messages', click the 'Export' button at the bottom to archive the messages as a file in CSV format. Thereafter, you can delete all the selected messages using the 'Delete' button at the bottom of the pate.