Added logs in MultiWriterIdGenerator.__init__() and get_cache_dict.

This issue has been migrated from [#16154](https://github.com/matrix-org/synapse/pull/16154).

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Fixes: https://github.com/matrix-org/synapse/issues/9443

* [x] Pull request is based on the develop branch
* [x] Pull request includes a [changelog file](https://matrix-org.github.io/synapse/latest/development/contributing_guide.html#changelog). The entry should:
- Be a short description of your change which makes sense to users. "Fixed a bug that prevented receiving messages from other servers." instead of "Moved X method from `EventStore` to `EventWorkerStore`.".
- Use markdown where necessary, mostly for `code blocks`.
- End with either a period (.) or an exclamation mark (!).
- Start with a capital letter.
- Feel free to credit yourself, by adding a sentence "Contributed by @github_username." or "Contributed by [Your Name]." to the end of the entry.
* [x] Pull request includes a [sign off](https://matrix-org.github.io/synapse/latest/development/contributing_guide.html#sign-off)
* [x] [Code style](https://matrix-org.github.io/synapse/latest/code_style.html) is correct
(run the [linters](https://matrix-org.github.io/synapse/latest/development/contributing_guide.html#run-the-linters))

Locate MultiWriterIdGenerator.__init__() and get_cache_dict in the Synapse codebase and review the surrounding logging conventions. Confirm what logging is needed in those entry points, then run the linters described in the contribution checklist. The issue is done when the requested logs are present and the checklist requirements remain satisfied.

Written by the indexing model from the issue text.