how can i log my program?

If you don't want to use syslog, just open a file in append mode and write your log data to it.

It's your decision where to store it. /var/log or a subdirectory are common places but your program requires write access to the directory.

For non-system programs you can create your program specific directory and write the log to that directory (or better a sub directory); e.g. /usr/local/myprogram/var/log. Or just use a subdirectory of your home dir if the program is only executed by you.

Ups. I missed that. Thank you for pointing to it.