There isn't an option to export logs from the appliance, but you can perform LEM database backups & archives to a Windows share. You can also use LEM Reports to export events to formats such as .txt, CSV, PDF and more. It is also possible to export nDepth search results to a CSV file.
Could you provide some information around the use case for exporting the logs?
If it is syslogs specifically that you are looking to export. The Ones you see in CMC > Appliance > Checklogs. You can use the exportsyslog command: Export log files using the CMC exportsyslog command - SolarWinds Worldwide, LLC. Help and Support
For the syslog dumps, what silverwolf says will work.
If you want to dump the LEM's internal database, you can do that with the backup commands described here: Configure Backups on your LEM Appliance - SolarWinds Worldwide, LLC. Help and Support