本番環境のlog/production.logにログが出力されず困っています。
開発環境では正常にログがlog/development.logに出力されます。

本番環境下でlog/production.logにログを出力させたいのでどなたかお教えください。

各環境の設定ファイルは以下のようになっております。

本番環境

# config/environments/production.rb
AppTool::Application.configure do
  config.cache_classes = true
  config.eager_load = false
  config.consider_all_requests_local       = false
  config.action_controller.perform_caching = true
  config.serve_static_assets = false
  config.static_cache_control = "public, max-age=3600"
  config.assets.js_compressor = :uglifier
  config.assets.compile = false
  config.assets.digest = true
  config.assets.version = '1.0'
  config.log_level = :info
  config.action_mailer.raise_delivery_errors = false
  config.i18n.fallbacks = true
  config.active_support.deprecation = :notify

  config.log_formatter = ::Logger::Formatter.new
  config.logger = Logger.new(config.paths["log"].first, 'weekly') # 週ごとにローテート
  config.logger.level = Logger::WARN
  config.logger.formatter = Logger::Formatter.new
  config.logger.datetime_format = "%Y-%m-%d %H:%M:%S"
end

開発環境

# config/environments/development.rb
AppTool::Application.configure do
  config.cache_classes = false
  config.eager_load = false
  config.consider_all_requests_local       = true
  config.action_controller.perform_caching = false
  config.action_mailer.raise_delivery_errors = false
  config.active_support.deprecation = :log
  config.active_record.migration_error = :page_load
  config.active_record.schema_format = :sql
  config.assets.debug = true

  config.logger = Logger.new(config.paths["log"].first, 'weekly') # 週ごとにローテート
  config.logger.level = Logger::DEBUG
  config.logger.formatter = Logger::Formatter.new
  config.logger.datetime_format = "%Y-%m-%d %H:%M:%S"
end

Rails.loggerの値をコマンドラインで確認すると以下のようになります。

本番環境
RAILS_ENV=production bundle exec rails c

irb(main):003:0* Rails.logger
=> #<Logger:0x007f70e1d8a5a0 @progname=nil, @level=1,
@default_formatter=#<Logger::Formatter:0x007f70e1d8a528
@datetime_format=nil>, @formatter=#
<ActiveSupport::Logger::SimpleFormatter:0x007f70e1d85500
@datetime_format=nil>, @logdev=#<Logger::LogDevice:0x007f70e1d8a4b0
@shift_size=nil, @shift_age=nil, @filename=nil, @dev=#<IO:<STDOUT>>,
@mutex=#<Logger::LogDevice::LogDeviceMutex:0x007f70e1d8a488
@mon_owner=nil, @mon_count=0, @mon_mutex=#<Mutex:0x007f70e1d8a438>>>>

開発環境
RAILS_ENV=development bundle exec rails c

[1] pry(main)> Rails.logger
=> #<Logger:0x007ff0eb129538
 @default_formatter=
  #<Logger::Formatter:0x007ff0eb129510 @datetime_format="%Y-%m-%d %H:%M:%S">,
 @formatter=#<Logger::Formatter:0x007ff0eb1292b8
 @datetime_format=nil>,
 @level=0,
 @logdev=#<Logger::LogDevice:0x007ff0eb129498
 @dev=#<File:/Users/xf-dev-mac02/apptool/log/development.log>,
 @filename="/Users/xf-dev-mac02/apptool/log/development.log",
 @mutex=#<Logger::LogDevice::LogDeviceMutex:0x007ff0eb129470
 @mon_count=0,
 @mon_mutex=#<Mutex:0x007ff0eb129420>,
 @mon_owner=nil>,
 @shift_age="weekly",
 @shift_size=1048576>,
 @progname=nil>

開発環境では@filenameの値が指定されるようになっています。