I’m betting the problem is that Cygwin is writing Unix line endings (LF) to the file, and you’re opening it with a program that expects Windows line-endings (CRLF). To determine if this is the case — and for a bit of a hackish workaround — try:
echo "`date` User `whoami` started the script."$'\r' >> output.log
(where the $'\r' at the end is an extra carriage-return; it, plus the Unix line ending, will result in a Windows line ending).