Typically, you would want to run certain background processes at start up. Let's take the log daemon, syslogd, as an example. The purpose of syslogd is to accumulate log messages from other programs, mostly other daemons. Naturally, BusyBox has an applet for that!
Starting the daemon is as simple as adding a line like this to etc/inittab:
::respawn:syslogd -n
respawn means that, if the program terminates, it will be automatically restarted; -n means that it should run as a foreground process. The log is written to /var/log/messages.
You may also want to start klogd in the same way: klogd sends kernel log messages to syslogd so that they can be logged to permanent storage.
As an aside, I should mention that, in the case of a typical embedded Linux system, writing log files to flash memory is not such a good idea as it will wear it out. I will cover the options for logging in Chapter 7, Creating a Storage Strategy.