Okay, this method should work regardless
of the amount of files you need to process.

Break On
Dim $SO
$SO=SetOption('Explicit','On')
$SO=SetOption('NoVarsInStrings','On')
$SO=SetOption('NoMacrosInStrings','On')

Dim $Path, $Dest, $File, $Files, $Folder, $Folders
$Path = 'C:\Office\Type'
$File = 'logfile.txt'
$Dest = 'C:\TEMP\AllUserLogFiles.txt'
If Exist($Dest)
Del $Dest
EndIf
$Folders = DirL($Path)
For Each $Folder In $Folders
If $Folder
If Exist($Path+'\'+$Folder+'\'+$File)
SHELL '%comspec% /c type ' + $Path+'\'+$Folder+'\'+$File + ' >>'+$Dest
EndIf
EndIf
Next

Function DirL($f)
Dim $a[0]
$f = Dir($f)
While Not @ERROR
If $f <> '..' And $f <> '.'
$a[UBound($a)] = $f
ReDim Preserve $a[UBound($a)+1]
EndIf
$f = Dir()
Loop
If UBound($a)
ReDim Preserve $a[UBound($a)-1]
Else
$a = 0
EndIf
$DirL = $a
EndFunction