If you want to extend java.util.logging.Logger you must create public constructor which call parent constructor with super.

public MyLogger(String name, String resourceBundleName) {
  super(name, resourceBundleName);
}

After that just use it to create MyLogger with it like

private final MyLogger log =
    new MyLogger(
        MyService.class.getPackageName() + "." + MyService.class.getName(), null);