By default, ansible.cfg has log_path commented out, like this.

#log_path = /var/log/ansible.log

When commented out, Ansible will log events to the machines syslog (system log daemon). As an example, syslog may write Ansible events to the /var/log/messages file.

Mar 22 12:55:08 [localhost] ansible-command: Invoked with creates=None executable=None _uses_shell=True strip_empty_ends=True _raw_params=ping -c4 was2.software.eng.us removes=None argv=None warn=True chdir=None stdin_add_newline=True stdin=None

To configure Ansible to log events to it's own log file, one option is to uncomment the log_path line. In this example, Ansible will start to log events to /var/log/ansible.log.

log_path = /var/log/ansible.log

Or you could set a temporary log path using the ANSIBLE_LOG_PATH variable.

export ANSIBLE_LOG_PATH=/home/john.doe/ansible.log

Create the log file and update the permission so that it has the write permission for user, group and other.

touch /var/log/ansible.log
chmod 0666 /var/log/ansible.log

You will probably also want to configure logrotate to rotate the log files.