User names in postgres docker image logs

Question

I'm looking to get thorough logs of user activity attributable back to users from the official docker image of Postgres. I'm running my docker image like so:

docker run --name pg -e POSTGRES_PASSWORD=password -p 54320:5432 postgres -c log_statement=all -c log_line_prefix=`t=%m u=%u db=%d pid=[%p]:

But I'm getting logs that look like:

PostgreSQL init process complete; ready for start up.

LOG:  starting PostgreSQL 13.1 (Debian 13.1-1.pgdg100+1) on x86_64-pc-linux-gnu, compiled by gcc (Debian 8.3.0-6) 8.3.0, 64-bit
LOG:  listening on IPv4 address "0.0.0.0", port 5432
LOG:  listening on IPv6 address "::", port 5432
LOG:  listening on Unix socket "/var/run/postgresql/.s.PGSQL.5432"
LOG:  database system was shut down at 2021-01-24 11:58:13 UTC
LOG:  database system is ready to accept connections
FATAL:  database "test" does not exist
LOG:  statement: SELECT VERSION()
LOG:  statement: SHOW ALL
LOG:  statement: SELECT datName FROM pg_database WHERE datAllowConn ORDER BY datName
LOG:  statement: CREATE DATABASE "test"
LOG:  statement: /* setup database */

    CREATE TABLE IF NOT EXISTS users (
       id SERIAL PRIMARY KEY,
       name varchar(255) NOT NULL,
       email varchar(255) NOT NULL,
       password_hash varchar(255) NOT NULL
    );

    /* fill with seed data */
    INSERT INTO users
    (name, email, password_hash) VALUES
    ('Robert', '[email protected]', 'hash'),
    ('Bob', '[email protected]', 'new_hash'),
    ('Rob', '[email protected]', 'another hash');
    /* read data, write data using good user */

    /* read data, write data */
LOG:  statement: SELECT name, setting FROM pg_settings WHERE source = 'session';

Which doesn't give me a good timestamp or attribute the activity to a user. What am I doing wrong?