Someone asked this on Reddit, to no conclusion.

In the source novel his logs are in text format, not video. At one point (after retrieving Pathfinder) he makes an assertion that he should be more careful with his logs,

now that someone might actually read them

However, ultimate fate of his logs is left undisclosed in both media.

And when you think about it, his logs only hold sentimental significance. Was he dead, NASA would try to rescue his logs (with his body) to collect information. But since they have a living Mark Watney, they could just ask whatever it is they're curios about (Potatoes? On Mars?!) directly to him.

Someone asked this on Reddit, to no conclusion.

In the source novel his logs are in text format, not video. At one point (after retrieving the Pathfinder) he makes an assertion that he should be more careful with his logs,

now that someone might actually read them

However, ultimate fate of his logs is left undisclosed in both media.

If you're asking about the films, then there is little chance he could take the logs with him. Logs seemed to be pretty HD (at least 720p), which would make them pretty bulky, disk space-wise. To take all of them home, he'd need a hard drive, which he couldn't store in his suit.

He did have USBs with crew's entertainment, but I imagine it would be insufficient to hold all of his logs.

And when you think about it, his logs only hold sentimental significance. Was he dead, NASA would try to rescue his logs (with his body) to collect information. But since they have a living Mark Watney, they could just ask whatever it is they're curios about (Potatoes? On Mars?!) directly to him (as pointed out by Andrew Thompson in comments).