log.cs

public static class DatabaseLogger
{
    public static void Log(string query)
    {
        var words = query.Split(' ');

        if (words.FirstOrDefault()?.ToUpper() == "UPDATE")
        {
            //aqui salva o log de update
        }
    }
}

//Lá no contexto 

public DatabaseContext : DbContext
{
    public DatabaseContext() : base("Connstring")
    {
        Database.Log = (message) => DatabaseLogger.Log(message);
    }
}
}