The answer to this question depends on which build is currently installed.
AutoUpdate Builds (builds after 2241)
All logs can be found in the logs directory in the qie.home directory.
In Windows 7, Windows 2008, Windows XP, the default location is here
C:\ProgramData\QIE\logs
In Windows 2003
C:\Documents and Settings\All Users\Application Data\QIE\logs
In Linux
/java/QIE/logs
Non AutoUpdate Builds (build 2241 and earlier)
The logs in these builds are split between the install directory and qieHome.
In Windows 7, Windows 2008, Windows XP, here is the default location
C:\Program Files (x86)\QIE\log
C:\ProgramData\QIE\qie.log
In Windows 2003
C:\Program Files (x86)\QIE\log
C:\Documents and Settings\All Users\Application Data\QIE\qie.log
In Linux
/java/QIE/qie.log