There is no public API for taking meeting logs for the new Teams that show meeting start and end times. To achieve your requirements, we recommend you give your feedback in Teams Feedback Portal.
However, there are Microsoft Graph APIs for retrieving Microsoft Teams meeting transcripts that are now generally available.
Ref: https://devblogs.microsoft.com/microsoft365dev/microsoft-graph-apis-for-microsoft-teams-meeting-transcripts-now-generally-available/
These APIs enable app developers to retrieve meeting transcripts programmatically for scheduled regular (non-channel) meetings.
Thanks,
Prasad Das
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