There are two methods to going about the audio logs on ODST for TMCC: getting all the logs as you progress through a run of the campaign, or simply getting them all in one run of the streets after you finish NMPD HQ (which coincides with Rally Point 6 on Mombasa Streets).
I highly advise the latter. Due to the glitchy nature (currently) of these logs not saving to your profile, ESPECIALLY if playing co-op, it is best to get them all in one go without quitting out and should only take about an hour.
Load up Rally Point 6 on Mombasa Streets (ignore the part in the video where I say to do NMPD HQ). You will be on Mombasa Streets with all the necessary paths open, and your next destination will be Kikowani Station.
The Rooster Teeth/Achievement Hunter map is great, but it is numbered more or less based on getting them as you progress. I elected to make a video that shows a more logical route to grabbing them all based on your starting location after NMPD HQ at this portion of Mombasa Streets, doing a loop of the map and ending at Kikowani.
You must collect all 29 logs in the streets, then carry on with the campaign. When you reach Data Hive, if you have all 29, a Marine will open a side door leading to the 30th one. Apparently if you're playing solo, from what I hear, you can quit out after getting all 29 and just load up Data Hive. Personally with how glitchy it is I'd just rush through Kikowani and all that just to be safe, but that's up to you.
I have not numbered them in the video because they dont have designated numbers. Whichever one you grab first will be Audio Log 1, the second will be log 2, etc regardless of what order you pick them up. Check out the video below for an on-foot walkthrough of the path I took to nab this.