Initializing Client interferes with application logging

A recent update of distributed seems to break my application's logging after a Client is initialized.

I have a minimal example attached:
[log-interference.zip](https://github.com/dask/distributed/files/2211796/log-interference.zip).
I set up a single file handler to capture both the distributed logs and the logs from my application. This works as expected in distributed v1.18.3; my application logs save to file before and after calling `Client()`

With the current version of distributed (1.22.0), note that the log "Started Client" is missing. In my actual application, all logs are missing after I call `Client()`.
There also happen to be some new logs from `asyncio` - I don't mind those, but do mind having my logs missing afterwards.

I don't know whether the difference comes from `distributed` or one of its dependencies. Here are my versions of all dependencies in the two environments I'm using: [versions.txt](https://github.com/dask/distributed/files/2211798/versions.txt)