Where does Applescript Event Log go when run under Launchd?
To bring the contents of Mateusz's comment here in an answer. Credit to Camelot for the steps.

The AppleScript log statement does not write to the StandardOutPath. Writing to a log file takes 3 steps. The second step may take 2 forms depending of wether you want to save previously written data.

-- Open the log file
set logFile to (open for access POSIX file "/var/tmp/myScript.log" with write permission)
-- Write some data to it (clearing the existing contents)
write "some data" &amp; return to logFile
-- Or, write by appending, some data to the end of existing data
write "some data" &amp; return to logFile starting at eof
-- Close the file
close access logFile


It may help to rewrite the 1st step as 2 statements:

set logFilePath to "/var/tmp/myScript.log"
set logFile to (open for access POSIX file logFilePath with write permission)


I am still searching for an answer regarding the log statements and event log. At this point, I think they go into deep space.