In one of Prof. Starnes' log entries he says that the civilization on Triacus was destroyed by a "natural catastrophe". Yet later when Spock is asked about them by Kirk, he says they were a "band of marauders" who "made constant war" and that they "were themselves destroyed by those they preyed upon".
When Spock grabs Kirk and they get into the turbolift, it starts up and goes exactly where they want it to go even though they don't give it any verbal commands or use the hand control.
No wonder McCoy doesn't like to use the transporter! Kirk beams down two security guards to the planet - they don't "lock on" to the planet or call up the guards already on the planet to get a lock-on signal, or do any kind of double-checking to see if there are any gravitational anomalies, or anything. Who would want to travel by a transporter when they're that careless?
The stardates of Starnes' entries are 5025.3, 5032.4, and 5038.3 But Kirk gives the stardate of when they arrive at the beginning of the episode as 5029.5, so apparently Starnes made two entries after he died!
Who is taking the picture of Professor Starnes as he records his entry? Presumably he's recording himself using his tricorder...but in the picture he's holding the tricorder.
How does Kirk know Gorgan is called Gorgan? The children never call him that, Spock never mentions it in the research. The first time anyone calls him Gorgan is...when Kirk calls him Gorgan.
What happened to the two security guards stuck on the planet? At the end of this episode they apparently just fly on to their next assignment.