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Actually, as I found in this answer here, and from Valorum's answer to this question. People surrounding this book and film can be pretty responsive to emails. So I emailed Andy Weir (I hope you don't mind me blacking out my name), but this was his reply:

So according to him, Mark uploads the logs and sends them using the Ares 4 Mav, although I have no recollection of that happening.

With regards to the movie...

We don't know
To elaborate, nowhere in the script from when Mark reaches the Schiaparelli Crater, to when the MAV takes off does he upload them or are they discussed.

INT. ROVER - ARES 4 MAV SITE - DAY
TITLE: Sol 538
Mark stares at the camera with a look that says, “Oh jesus these JPL guys are gonna get me killed.”

[...]

EXT. MAV - DAY
Mark stands at the base of the MAV. He holds a large wrench in his hand, almost like a weapon.
As Mark stares up at the MAV with a gleam in his eyes...

INT. MAV - DAY
Mark tears the acceleration chairs out of the cockpit.

EXT. MAV - DAY
WHUMP. One after another, the acceleration chairs hit the dirt in a pile. WHUMP.

INT. MAV - DAY
Mark tears out the control panels. He’s having fun.

EXT. MAV - DAY
WHUMP. The controls hit the dirt. The pile is growing.

INT. MAV COCKPIT - DAY
Mark waits in the airlock with a mess of stripped equipment.

EXT. MAV - DAY
The outside of the MAV now looks like the set of Sanford and Son. Mark wrenches one of the MAV’s hull panels free.

EXT. MAV - DAY
UP ABOVE: The nose airlock breaks free, and tumbles down towards camera, BLACKING OUT FRAME.

EXT. MAV - DUSK
FROM BLACK, we FIND MARK. He’s sitting on a hill slope, surveying his handiwork.
The MAV has been TRANSFORMED. The whole front has been torn off. Hab canvas now covers it. Equipment litters the area all around us. Junkyard on Mars.
Mark just sits. Exhausted.

TITLE: Sol 560
INT. POP UP TENT - DAY
Mark sits inside the makeshift pressurized tent. He tears open his last remaining ration pack: “Goodbye, Mars”
He eats in silence.

After that section the only mention of Mark on Mars is strapped into the MAV waiting to launch. As Santa said, the book uses written logs whereas the film uses video logs, and as Andy Weir's email references the book, we have to look at the film separately. However, Andy Weir tells us that the MAVs had high-bandwidth connection which would allow for video transmission nearly as well as text transmission, so the best we can do is to assume that Ridley Scott and Drew Goddard had similar intentions to Andy Weir, having Mark Watney send them back to Earth via the Ares 4 MAV.

Actually, as I found in this answer here, and from Valorum's answer to this question. People surrounding this book and film can be pretty responsive to emails. So I emailed Andy Weir (I hope you don't mind me blacking out my name), but this was his reply:

So according to him, Mark uploads the logs and sends them using the Ares 4 Mav, although I have no recollection of that happening.

With regards to the movie...

We don't know
To elaborate, nowhere in the script from when Mark reaches the Schiaparelli Crater, to when the MAV takes off does he upload them or are they discussed.

INT. ROVER - ARES 4 MAV SITE - DAY
TITLE: Sol 538
Mark stares at the camera with a look that says, “Oh jesus these JPL guys are gonna get me killed.”

[...]

EXT. MAV - DAY
Mark stands at the base of the MAV. He holds a large wrench in his hand, almost like a weapon.
As Mark stares up at the MAV with a gleam in his eyes...

INT. MAV - DAY
Mark tears the acceleration chairs out of the cockpit.

EXT. MAV - DAY
WHUMP. One after another, the acceleration chairs hit the dirt in a pile. WHUMP.

INT. MAV - DAY
Mark tears out the control panels. He’s having fun.

EXT. MAV - DAY
WHUMP. The controls hit the dirt. The pile is growing.

INT. MAV COCKPIT - DAY
Mark waits in the airlock with a mess of stripped equipment.

EXT. MAV - DAY
The outside of the MAV now looks like the set of Sanford and Son. Mark wrenches one of the MAV’s hull panels free.

EXT. MAV - DAY
UP ABOVE: The nose airlock breaks free, and tumbles down towards camera, BLACKING OUT FRAME.

EXT. MAV - DUSK
FROM BLACK, we FIND MARK. He’s sitting on a hill slope, surveying his handiwork.
The MAV has been TRANSFORMED. The whole front has been torn off. Hab canvas now covers it. Equipment litters the area all around us. Junkyard on Mars.
Mark just sits. Exhausted.

TITLE: Sol 560
INT. POP UP TENT - DAY
Mark sits inside the makeshift pressurized tent. He tears open his last remaining ration pack: “Goodbye, Mars”
He eats in silence.

After that section the only mention of Mark on Mars is strapped into the MAV waiting to launch. As Santa said, the book uses written logs whereas the film uses video logs, and as Andy Weir's email references the book, we have to look at the film separately. However, Andy Weir tells us that the MAVs had high-bandwidth connection which would allow for video transmission nearly as well as text transmission, so the best we can do is to assume that Ridley Scott and Drew Goddard had similar intentions to Andy Weir, having Mark Watney send them back to Earth via the Ares 4 MAV.

INT. ROVER - ARES 4 MAV SITE - DAY
TITLE: Sol 538
Mark stares at the camera with a look that says, “Oh jesus these JPL guys are gonna get me killed.”
MARK (distraught)
I know what they’re doing. I know what they’re doing. They keep repeating “accelerate faster than any man in the history of space travel” like this is a good thing, like this’ll distract me from how insane their plan is. Oh really? I get to be the fastest man in the history of space travel? You’re launching me into space in a convertible. No no, it’s worse, because I don’t have any controls. You’re launching me into space in a tin can. And, by the way, physicists don’t even use words like “fast” when describing acceleration, so they’re only doing it in hopes I won’t raise any objections because I like the way “fastest man in the history of space travel” sounds. Well, you know what?
(thinks about it)
I do like it. I do like the way it sounds.
(then)
Okay, fine. Let’s do this.

INT. ROVER - ARES 4 MAV SITE - DAY
TITLE: Sol 538
Mark stares at the camera with a look that says, “Oh jesus these JPL guys are gonna get me killed.”

[...]

With regards to the movie...

We don't know
To elaborate, nowhere in the script from when Mark reaches the Schiaparelli Crater, to when the MAV takes off does he upload them or are they discussed.

INT. ROVER - ARES 4 MAV SITE - DAY
TITLE: Sol 538
Mark stares at the camera with a look that says, “Oh jesus these JPL guys are gonna get me killed.”
MARK (distraught)
I know what they’re doing. I know what they’re doing. They keep repeating “accelerate faster than any man in the history of space travel” like this is a good thing, like this’ll distract me from how insane their plan is. Oh really? I get to be the fastest man in the history of space travel? You’re launching me into space in a convertible. No no, it’s worse, because I don’t have any controls. You’re launching me into space in a tin can. And, by the way, physicists don’t even use words like “fast” when describing acceleration, so they’re only doing it in hopes I won’t raise any objections because I like the way “fastest man in the history of space travel” sounds. Well, you know what?
(thinks about it)
I do like it. I do like the way it sounds.
(then)
Okay, fine. Let’s do this.

EXT. MAV - DAY
Mark stands at the base of the MAV. He holds a large wrench in his hand, almost like a weapon.
As Mark stares up at the MAV with a gleam in his eyes...

INT. MAV - DAY
Mark tears the acceleration chairs out of the cockpit.

EXT. MAV - DAY
WHUMP. One after another, the acceleration chairs hit the dirt in a pile. WHUMP.

INT. MAV - DAY
Mark tears out the control panels. He’s having fun.

EXT. MAV - DAY
WHUMP. The controls hit the dirt. The pile is growing.

INT. MAV COCKPIT - DAY
Mark waits in the airlock with a mess of stripped equipment.

EXT. MAV - DAY
The outside of the MAV now looks like the set of Sanford and Son. Mark wrenches one of the MAV’s hull panels free.

EXT. MAV - DAY
UP ABOVE: The nose airlock breaks free, and tumbles down towards camera, BLACKING OUT FRAME.

EXT. MAV - DUSK
FROM BLACK, we FIND MARK. He’s sitting on a hill slope, surveying his handiwork.
The MAV has been TRANSFORMED. The whole front has been torn off. Hab canvas now covers it. Equipment litters the area all around us. Junkyard on Mars.
Mark just sits. Exhausted.

TITLE: Sol 560
INT. POP UP TENT - DAY
Mark sits inside the makeshift pressurized tent. He tears open his last remaining ration pack: “Goodbye, Mars”
He eats in silence.

After that section the only mention of Mark on Mars is strapped into the MAV waiting to launch. As Santa said, the book uses written logs whereas the film uses video logs, and as Andy Weir's email references the book, we have to look at the film separately. However, Andy Weir tells us that the MAVs had high-bandwidth connection which would allow for video transmission nearly as well as text transmission, so the best we can do is to assume that Ridley Scott and Drew Goddard had similar intentions to Andy Weir, having Mark Watney send them back to Earth via the Ares 4 MAV.