@robcool Welcome to Microsoft Q&A Forum, Thank you for posting your query here!
May I know, how are you trying to change the access level of "$log" folder (Portal, or Storage Explorer tool) ?
Can you please share the screen shot of the error message? Please try to change the access level through storage explorer tool and let me know the status
All logs are stored in block blobs in a container named $logs, which is automatically created when Storage Analytics is enabled for a storage account. The $logs container is located in the blob namespace of the storage account, for example: http://<accountname>.blob.core.windows.net/$logs. This container cannot be deleted once Storage Analytics has been enabled, though its contents can be deleted. If you use your storage-browsing tool to navigate to the container directly, you will see all the blobs that contain your logging data.
Assign an Azure role to access blob data
Please let us know if you have any further queries. I’m happy to assist you further.
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