I would suggest putting the log file onto a seperate disk, though should give you a little performance gain so that your not trying to both read and write to the same disk as the website. If you cannot put the log file on a seperate disk, then I would simply choose a folder of your choice.
In any case, you will have to give the "Network Service" account "Modify" permissions to the desired folder.
If on the other hand, you have access to a databse, then log the information there. It will be much quicker than accessing the hard drive and won't be publically available. You'll also be able to report from the data quite easily.