Ubuntuのdocker-composeでmysqlを立てるときのパーミッション設定

WindowsやMacOSで使うDocker Desktopとはパーミッション周りの挙動が少し違うみたいです。
my.cnfでログ出力設定をした際にエラーが出ました。

動くdocker-compose.yml

version: "3.7"

services:
  mysql:
    image: mysql:8
    volumes:
      - db_data:/var/lib/mysql
      - type: bind
        source: "./my.cnf"
        target: "/etc/mysql/conf.d/my.cnf"
    environment:
      MYSQL_ROOT_PASSWORD: ${MYSQL_ROOT_PASSWORD}
      TZ: ${TZ:-UTC}
    command: >
      bash -c "mkdir -p /var/log/mysql
      && chown -hR mysql:mysql /var/log/mysql
      && /entrypoint.sh mysqld"
    ports:
      - ${MYSQL_PORT:-3306}:3306
    restart: unless-stopped

volumes:
  db_data:

※ commandの３行はインデントがずれると動かない場合があるので注意

追記部分

+    command: >
+      bash -c "mkdir -p /var/log/mysql
+      && chown -hR mysql:mysql /var/log/mysql
+      && /entrypoint.sh mysqld"

エラーと対処

No such file or directory

mysql_1       | 2022-06-30 01:50:51+00:00 [Note] [Entrypoint]: Entrypoint script for MySQL Server 8.0.29-1debian10 started.
mysql_1       | 2022-06-30 01:50:51+00:00 [Note] [Entrypoint]: Switching to dedicated user 'mysql'
mysql_1       | 2022-06-30 01:50:51+00:00 [Note] [Entrypoint]: Entrypoint script for MySQL Server 8.0.29-1debian10 started.
mysql_1       | 2022-06-30 01:50:51+00:00 [Note] [Entrypoint]: Initializing database files
mysql_1       | 2022-06-30T01:50:51.803532Z 0 [System] [MY-013169] [Server] /usr/sbin/mysqld (mysqld 8.0.29) initializing of server in progress as process 41
mysql_1       | 2022-06-30T01:50:51.805709Z 0 [ERROR] [MY-010187] [Server] Could not open file '/var/log/mysql/error.log' for error logging: No such file or directory
mysql_1       | 2022-06-30T01:50:51.805753Z 0 [ERROR] [MY-013236] [Server] The designated data directory /var/lib/mysql/ is unusable. You can remove all files that the server added to it.
mysql_1       | 2022-06-30T01:50:51.805762Z 0 [ERROR] [MY-010119] [Server] Aborting

→ ディレクトリ作成

mkdir -p /var/log/mysql

Permission denied

mysql_1       | 2022-06-30 01:56:49+00:00 [Note] [Entrypoint]: Entrypoint script for MySQL Server 8.0.29-1debian10 started.
mysql_1       | 2022-06-30 01:56:49+00:00 [Note] [Entrypoint]: Switching to dedicated user 'mysql'
mysql_1       | 2022-06-30 01:56:49+00:00 [Note] [Entrypoint]: Entrypoint script for MySQL Server 8.0.29-1debian10 started.
mysql_1       | 2022-06-30 01:56:49+00:00 [Note] [Entrypoint]: Initializing database files
mysql_1       | 2022-06-30T01:56:49.795239Z 0 [System] [MY-013169] [Server] /usr/sbin/mysqld (mysqld 8.0.29) initializing of server in progress as process 44
mysql_1       | 2022-06-30T01:56:49.796978Z 0 [ERROR] [MY-010187] [Server] Could not open file '/var/log/mysql/error.log' for error logging: Permission denied
mysql_1       | 2022-06-30T01:56:49.797012Z 0 [ERROR] [MY-013236] [Server] The designated data directory /var/lib/mysql/ is unusable. You can remove all files that the server added to it.
mysql_1       | 2022-06-30T01:56:49.797019Z 0 [ERROR] [MY-010119] [Server] Aborting

→ パーミッション設定

chown -hR mysql:mysql /var/log/mysql