This article explains if Action logs can be streamed to a Log Stream.

• Actions

As of now, the output of console.log statements in Rules, Actions, and other extensibility points is not part of the tenant log data exported via Log Streams. These runtime logs are only viewable internally, mainly by using the Realtime Webtask Logs extension, although the output can be seen in the 'Actions Details' tab of a tenant log that has triggered an Action.

The logs exported via Log Streams are all the JSON data included under the 'Raw' tab in the Details section of a tenant log (in the Management dashboard).

As a workaround, an action could send log information to an external service using some webhook.
Create a custom API, invoke this API through Actions, and send logs there.

See Store the Auth0 user id in a remote system for an example of how to use Axios to call an external API