For a standard installation (ie non-portable) the folder paths, including Index Log folder, are read from the registry value:
HKEY_CURRENT_USER\Software\Mythicsoft\FileLocatorPro\Core\Folders
For portable installs the paths are loaded from the file:
master.xml
The paths are stored as XML and will look something like:
<cfg ver="2">
<section name="FOLDERS">
<LogFile t="3">$(ApplicationData)\logs\</LogFile>
<CacheFolder t="3">$(ApplicationData)\Cache\</CacheFolder>
<ConfigFile t="3">$(ApplicationData)\config\</ConfigFile>
<IndexLogFolder t="3">$(ApplicationData)\IndexLog\</IndexLogFolder>
<FavouriteFolder t="3">$(ApplicationData)\Favorites\</FavouriteFolder>
<IndexFolder t="3">$(ApplicationData)\Index\</IndexFolder>
<CrashReports t="3">$(ApplicationData)\CrashReports\</CrashReports>
<DefaultData t="3">$(ApplicationData)</DefaultData>
<TempFolder t="3">$(TempFolder)\</TempFolder>
</section>
</cfg>
Change the IndexLogFolder element to change the default path for the Index Logs.
More information
The Index Log folder stores the log database, which contains information about which files were indexed and any issues encountered. The Index Manager tabs Update Log, History, and Messages and loaded from the Index Log.

Help: Index Details
What happens if I delete the log?
Index searching does not use the Index Log so you can delete the log database without affecting searches.