When you collect the Spartan Audio Logs in Ringfall, they only unlock in numerical order. The important part in getting the Spartan Audio Logs are the location of each Audio Log. Here is a guide on how to get all 8 Ringfall Spartan Audio Logs.
Ringfall Spartan Audio Log Locations
Go to the southeast of Balkarus to find the 1st Audio Log on top of the pillars. Head to the right of Fob Charlie to get the 2nd Audio Log next to the rock on the mountain. The 3rd Audio Log is to the west of Thunder Squad, inside the crashed vehicle.
Head to the southwest of the Forge of Teash until you reach a lake. You will see the 4th Audio Log on top of the pillar. From here, go southwest and up the mountains to get the 5th Audio Log on top of the rock.
The 6th Audio Log is near the Pyramid Squad, next to a dead soldier. Go to the very bottom of the map and you will find the 7th Audio Log next to a crate. The last Audio Log is northeast of Fob Echo, next to another dead soldier.