I set up a task using Import data with custom request and ran it. It failed the first time, reporting that it couldn't find the log file. Interestingly enough, it reported that issue in the very same log file it said it couldn't find! :-)
I have my data integration job in a sub folder (Contacts in the above image) which is also reflected in the log file location. Maybe on the first log attempt that folder didn't exist yet?
