make a log file
The standard way to log from a C program is syslog.
Start by including the header file:
#include <syslog.h>
Then early in your program, you should configure syslog by calling openlog:
openlog("programname", 0, LOG_USER);
The first argument is the identification or the tag, which is automatically added at the start of each message. Put your program's name here.
The second argument is the options you want to use, or 0 for the normal behavior. The full list of options is in man 3 syslog. One you might find useful is LOG_PID, which makes syslog also record the process id in the log message.
Then, each time you want to write a log message, you call syslog:
syslog(LOG_INFO, "%s", "Message");
The first argument is the priority. The priority ranges from DEBUG (least important) to EMERG (only for emergencies) with DEBUG, INFO, and ERR being the most commonly used. See man 3 syslog for your options.
The second and third arguments are a format and a message, just like printf.
Which log file this appears in depends on your syslog settings.
With a default setup, it probably goes into /var/log/messages.
You can set up a custom log file by using one of the facilities in the range LOG_LOCAL0 to LOG_LOCAL7.
You use them by changing:
openlog("programname", 0, LOG_USER);
to
openlog("programname", 0, LOG_LOCAL0);
or
openlog("programname", 0, LOG_LOCAL1);
etc.
and adding a corresponding entry to /etc/syslog.conf, e.g.
local1.info /var/log/programname.log
and restarting the syslog server, e.g.
pkill -HUP syslogd
The .info part of local1.info above means that all messages that are INFO or more important will be logged, including INFO, NOTICE, ERR (error), CRIT (critical), etc., but not DEBUG.
Or, if you have rsyslog, you could try a property-based filter, e.g.
:syslogtag, isequal, "programname:" /var/log/programname.log
The syslogtag should contain a ":".
Or, if you are planning on distributing your software to other people, it's probably not a good idea to rely on using LOG_LOCAL or an rsyslog filter.
In that case, you should use LOG_USER (if it's a normal program) or LOG_DAEMON (if it's a server), write your startup messages and error messages using syslog, but write all of your log messages to a file outside of syslog. For example, Apache HTTPd logs to /var/log/apache2/* or /var/log/httpd/*, I assume using regular open/fopen and write/printf calls.
You will want to #include <syslog.h>, then use the syslog() functions to send data to whatever system logging program is active.
See the man page here.