Possibly Call Me Mr. Positive by Tom Barlow. I read this in the anthology Orson Scott Card's InterGalactic Medicine Show edited by Edmund R. Schubert and Orson Scott Card. According to IFSDB it has only appeared in that anthology.
We never learn the protagonist's name, or the name of the ship and the details of the voyage it's on. All we know that they have been woken as it's their turn to man the ship:
It was my watch ... This was my ninth awake period of the voyage,
The protagonist does a routine check of the other crew members and finds:
As soon as I saw the first body, I knew the rest would be dead. The readouts were there in plain sight, right in front of me when I woke up, but I hadn’t bothered to look. I had just assumed everything was all right. Things couldn’t be any less right.
I checked them over one at a time anyway. Every one hurt just as much as the first, or maybe more.
They weren’t smashed-faceplate dead. They were peaceful-sleep dead. They looked like they’d died at about the same time, and not too long ago; there wasn’t a great deal of decomposition
The story isn't the protagonist reading the log, but rather writing entries in the log. The story is a series of log entries.
At the end of the story the protagonist returns to his pod:
I hadn’t admitted it to myself yet, but I was finally ready to get in the pod. Afraid my courage would evaporate if I looked at it too carefully, I let my mind go blank as I dressed and prepped the pod. I slid in, and was about to close the lid, but I couldn’t shake the notion that I’d left something unfinished.
I got up and wrote these words so that the log has some sort of an ending, in case things don’t turn out well. I’ve always hated books that end “To be continued.”
I thought long and hard about these, perhaps my last words. I was looking for something profound, something you could carve on my gravestone if you want to, but couldn’t think of anything. Only that I’d rather be floating dead through space with five of my friends than be alive and alone.
See you in six months. Call me Mr. Positive.
You know what’s funny? The cabin? It has a night light.