There is no such log, at least not by default.
There are tools you can install, like
acct ("GNU Accounting utilities for process and login accounting") which can keep track of all programs executed on the system.
acct is packaged for most (probably all except tiny-distros) linux distros. The home page is at http://www.gnu.org/software/acct/
acct provides the following commands,
lastcomm probably does what you're asking for:
prints statistics about users' connect time. ac can tell you how long a particular user or group of users were connected to your
system, printing totals by day or for all of the entries in the wtmp
turns accounting on or off.
lists the commands executed on the system, most recent first, showing the run state of each command. With last, you can search the
acct file for a particular user, terminal, or command.
summarizes the information in the acct file into the savacct and usracct file. It also generates reports about commands, giving the
number of invocations, cpu time used, average core usage, etc.
display acct and utmp files in a human-readable format.