Mac OS X: On Mac OS X, it's written to your Pictures/VueScan folder (before 9.5.84 it's written to your Library/Preferences folder). You can find this file with Spotlight in 9.5.84 and later.
Windows: On Windows, it's written to your Pictures\VueScan folder (or your My Pictures\VueScan folder on Windows XP).
Linux: On Linux, it's written to your ~/.vuescan folder (the period at the front makes it a hidden folder).
When you e-mail vuescan.log, send it as a file attachment, not text in the body of the e-mail. Please don't send it in .zip, .pdf, .doc, .gz, .7z, or any other compressed format - just vuescan.log.
The vuescan.log file is always written to your hard drive, even if VueScan hangs when you run it.