When you complete the first chapter of the game, you will find yourself aboard Sanctuary III. There you will meet up with several familiar faces. Throughout the ship, there are a few hidden echo logs. You can find Amara‘s echolog in Moxxi’s bar. When you enter the establishment the recording is hidden behind a Torgue cereal box on one of the tables on the right side of the bar.
I like how these data disks are like old NES cartridges. From time to time your Vault Hunter will blow into the cartridge in order to get them working.