If you use the FOR command to list files, it never redirects the output to a log file because the echo command will never be executed if the FOR command does not iterate the file names.
@echo off
set "LOGFILE=output.txt"
del "%logfile%"
:LOOP
set /P "userInputPath=Enter the path you'd like to search;"
set /p "FileType=Enter file type(s) here (ex: txt, pdf, docx):"
IF NOT EXIST "%userInputPath%" (
echo %userInputPath% does not exist
GOTO LOOP
)
for /R "%userInputPath%" %%G in (%FileType%) do echo %%G>>%LOGFILE%
pause