There are 40 Audio Logs
Audio Log #21 - During the "Find the Cultist Reinforcements" objective, the log is on the ground behind a tower with a few enemies. If you get the "Eliminate the Cultists" objective, turn around.
Audio Log #22
Audio Log #23 - On the body of a dude halfway up during the Save the Terraformer objective
Audio Log #33 - When you reach the bottom of the tomb, shortly before the first upgrade station.
Audio Log #24 - Outside of a trailer with stairs when you're in the caves
Audio Log #02 - At the beginning of the Get to the Mining Colony objective in the caves to the right in a fiery tunnel
Audio Log #08 - After Clearing the Area, and before Activating the Elevator, look around the large open area. You'll find this log outside of a tunnel entrance.
Audio Log #36 - After you get off the elevator to the left past the assault rifle
Audio Log #11 - To the left of the Excavator, near the Upgrade Bench
Audio Log #25 - To the right down a hall in the room with the big green enemies during the mission objective Get to the Gondola
Note: You must exit the Exo to pick up an audio log
Audio Log #01 - After you pick up the second power cell, to the right of the hall with the turrets pointing down it.
Audio Log #05 - Next to the generator at the top of the stairs that you have to repair.
Audio Log #27 - Under a half cylinder roof building by where Sam gets killed by the green monster, right before you enter Bastion.
Audio Log #37 - In a shack to the right at the beginning of the Find the Water Pumps objective
Audio Log #28 - After repairing the Water Pumps, you'll find an Upgrade Bench next to a chasm. Sprint and jump over the chasm.
Audio Log #19 - At the back of the room right after you get the Nanite Gun
Audio Log #15 - To the right just after you pick up the Rocket Launcher.
Audio Log #16 - During the Find the Corporal objective, there's a fork in the road. Go down the stairs instead of up towards the objective.
Log #17 - During the "Get to the Purification Plant" objective, take the upper platform walkway, and there's a hidden room on the left.
Log #13 - During the "Destroy the Infected Buildings" objective, between the 3rd and 4th buildings you have to destroy.
Log #14 - Near the last build you have to destroy
Log #31 - During the Meet Kara at the Market objective, it's next to a Weapons Locker
Log #31 - During the Find Kara objective, it's in a small room on the left. You'll have to get out of the Exo to pick it up
Log #29 - At the top of the ladder you're forced to climb, after getting out of the Exo
Log #32 - During the Find and Destroy Alien Pods objective, it's at the top of a hill.
Log #07 - Right by the Rail Driver
Log #34 - During the Destroy the Relay Stations objective, it's near one of the towers you must destroy at the top of a hill.
Log #03 - During the Get to the Front Line objective, it's on a small platform over lava.
Log #04 - During the Get to the Bridge objective, it's right before you start the Defend the Area objective
Log #18 - During the Find the Lab objective, right next to a weapon locker, impossible to miss.
Log #09 - During the Find the Lab objective, you'll walk along a narrow walkway. After exiting the walkway, it's in a room on the left.
Log #10 - During the Find the Lab objective, you'll walk down a long narrow ramp, and it's on the main path on the left.