In the far reaches of Argon territory you will find a space station that's NOT a factory.  It's the Goner temple.  When you contact them, they will tell you about the secret log of their (and your) hero Dunne.
a. The log is hidden in an asteroid.
b. The asteroid is in Chin's Cloud sector.
c. You can recognize it because a mineral scanner will show an unknown mineral.
d. If you find it, and blow it up, you can recover Dunne's log. Afterward, come back and talk to them again.
When you get the log, you'll have to pick it up as cargo.  But it won't show on your Ship screen.  It will show in your Pilot's log, however.
When/after you get the log, you'll be told to report to the Argon Destroyer. When you return to the Goners, they tell you the same thing.