Unix has had for a long while a special logging framework called syslog. Type in your shell
man 3 syslog
and you’ll get the help for the C interface to it.
Some examples
#include <stdio.h>
#include <unistd.h>
#include <syslog.h>
int main(void) {
openlog("slog", LOG_PID|LOG_CONS, LOG_USER);
syslog(LOG_INFO, "A different kind of Hello world ... ");
closelog();
return 0;
}