Captain's log, USS Discovery
The captain's log on the  was the method used by the  and acting Captain to officially record the ship's missions.

Entries

2256 
 "Captain's log, stardate 2137.2. Captain Harcourt Fenton Mudd recording. I am about to close the deal of the millennium." ()

2257 
 "Captain's log, stardate 1834.2. Acting Captain Saru recording. We are en route to the Terran flagship, having been summoned by Specialist Burnham. Discoverys spore drive is operational again, thanks to the full recovery of our navigator, Lieutenant Stamets. Yet, despite his best efforts, the lieutenant was unable to save Discoverys mycelium crop. His fear is that his   has already contaminated the network, and if so, the results could be catastrophic." ()

3189 
"Captain’s log, supplemental. After a journey of 930 years and a return to Earth that we could not have anticipated, we are finally on the verge of reaching our destination: Federation and Starfleet headquarters. Separate entities that must now abide together, a sign of this new time, I suppose. I can only hope that they are as eager for us to arrive as we are to be home." ()

3190 
"Captain's log, stardate 865661.2. It's been a week since my mission with the Qowat Milat. Since then, we've stayed in orbit above Ni'Var, working with the Ni'Var Science Institute and the Federation task force that's tracking and studying the DMA. No other inhabited worlds have been threatened yet, but of course that can change at any moment. Ni'Var has fast-tracked negotiations to rejoin the Federation. I haven't yet heard how they're going. We're all living in uncertainty. Even for a crew as familiar with the unknown as this one, the stress is taking its toll. I'm following Dr. Culber's advice, mandating downtime to help with their psychological and emotional well-being. But Stamets won't let himself slow down, even for a second. He wants to solve this for all of us, especially Book. Book. Even though the mind meld with T'Rina helped at the time, the peace he felt has been… fleeting. I've encouraged him to talk to Dr. Culber, but I feel him pulling into himself. A natural response to grief, crisis, all of this. But he can't do this alone. None of us can." ()

"Captain's log, stardate 865783.7. We have less than 29 hours before United Earth and Ni'Var start getting hit by debris from the DMA. 29 hours, billions of lives. Until I figure out how to get us in front of the Ten-C, there is nothing we can do to stop this. They live inside of a hyperfield, and right now I have no idea how to get in, what we'll find once we do, how to communicate with them, what they even look like. As a xenoanthropologist, I've never had less to work with. And as a captain, I have never been more afraid. Our only hope is the dead planet we're orbiting right now. It was a gas giant until a series of massive asteroids hit and burned the gas away. Its star is surrounded by Dyson rings made from the same rare material as the DMA controller, and my gut tells me there has to be a connection. That this planet must have been the Ten-C's home at some point. And the ruins down there can tell us how they lived, what's important to them. Cultural context, a way to begin communicating. If I'm wrong, we'll have no way to make up the time we spend down there. And the DMA will be that much closer to our homes. But if I'm right, and something we find can help us connect with the Ten-C, then maybe we have a shot at stopping this before it's too late. I have to be right. I have to." ()

Background information
A deleted scene from  had at least one log entry that was made in 2256 by Commander Saru captain's personal log.

"Acting captain's personal log. Stardate 4774.7.  Awaiting Captain 's return from the War Council with news of our next mission. Though it has been a routine day, I am ill at ease. Vigilance equals survival. As such, I have order continuous scans searching the darkness for all things dubious, dangerous, worthy of fear. There is nothing except the unmapped, starless planet we orbit. I have christened it "The Orb of Our Undoing"."

Discovery 00