No. - if you are exporting to a pst from Outlook there is no logging of that.
If you are doing this from an Exchange Server using New-MailboxExportRequest, it will be logged in the Admin Audit Logs
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On-premise Exchange server
Folks - If I backup my inbox to a local PST file, does the Exchange server administrator get notified?
Thank you
No. - if you are exporting to a pst from Outlook there is no logging of that.
If you are doing this from an Exchange Server using New-MailboxExportRequest, it will be logged in the Admin Audit Logs