In Unix based systems, is there a
log file that stores user's executed command(s)?
Given that you want to track all user commands, you should look at the
acct package on your system (on some systems this is also called "process accounting" or
psacct). Then after it's been turned on, you can run the
lastcomm command to show what programs have been run, by whom, when and for how long. From Google, search "linux acct" for more details.