In its default config, HAProxy send logs to a syslog facility: local0, via a socket connection. By default, your syslog configuration probably doesn’t accept socket connections, and even doesn’t have a local0 facility, so you have no HAProxy log. If you want it, configure syslog to accept TCP connections by adding -r to syslogd parameters.

On a RedHat/CentOS server, edit the value of SYSLOGD_OPTIONS in /etc/sysconfig/syslog:

SYSLOGD_OPTIONS="-m 0 -r"

On a Debian/Ubuntu server, edit the value of SYSLOGD in /etc/default/syslogd:

SYSLOGD="-r"

Then set up syslog facility local0 and direct it to file /var/log/haproxy.log or your desired location by editing /etc/syslog.conf:

local0.* /var/log/haproxy.log

finally, restart syslog service to apply changes:

$ /etc/init.d/syslog restart

You can see that HAProxy logging is started.

Other useful article on this blog that you may like to read:
* Configure HAProxy for content switching
* 5 Steps to secure your Production Linux Server
* Ensuring secure access to your Production Linux Server
* Bash script to backup essential log files in Linux Server