Why does mysql is not creating binlog?

Issue

• Mysql is not creating binlog with the following my.cnf configuration. What is wrong?

  [mysqld]
  skip-locking
  skip-name-resolve
  skip-slave-start
  datadir=/var/lib/mysql
  socket=/var/lib/mysql/mysql.sock
  user=mysql
  thread_cache_size = 256
  table_cache = 1024
  key_buffer = 64M
  sort_buffer_size = 256K
  read_buffer_size = 256K
  read_rnd_buffer_size = 256K
  max_allowed_packet = 10M
  tmp_table_size=32M
  max_heap_table_size=32M
  query_cache_size=128M
  query_cache_type=1
  log_output=FILE
  slow_query_log_file=/var/lib/mysql/slow-query.log
  slow_query_log=1
  long_query_time=2
  symbolic-links=0
  server-id       =2
  
  [mysqld_safe]
  log-error=/var/log/mysqld.log
  pid-file=/var/run/mysqld/mysqld.pid
  log-bin=mysql-log