When Goku grappled with Vegeta (Saiyan saga)...

[Desassina]

... after he used Kaioken x3, do you think Goku was still under its influence? Or did Vegeta's power lower so much (but not below Goku's) that Galick Gun was 2-3x the increase as well? I was trying to describe the battle with numbers, as if I could recreate it in my mind without watching it, but I was left in doubt when it came to this moment. I'll show you why:

(I recommend you pick up a calculator, because when I say that he is 0.75 times less than another number, I want to multiply it by an endless one such as 1.33...)

Goku - 8'000 -> 16'000 - he uses Kaioken x2 to bring out power that he can control, and whatever loss in power is quickly recovered, to make things simpler.
Vegeta - 18'000 - after he shows the true power of a Saiyan. Even if he took hits, they were not as damaging to bring his power down, which increased to dodge KKx2.

Goku - 8'000 -> 24'000 - he uses Kaioken x3, which is 1.5 times higher than KKx2, so I'll use it to describe his power loss. Goku's power decreases to 5'333.
Vegeta - 18'000 -> (6'000)? - Vegeta could have been damaged by twice the amount of Goku's increase in power, which brings us to 3 times less than his original power.

Goku - 5'333 -> 16'000 -> 21'333 - he uses Kaioken x3 to match Vegeta's GG, and then goes over to KKx4, and because it's 0.75 times less than before, Goku is now 4'000.
Vegeta - 6'000 -> 16'000 - he uses Galick Gun, which could have been 0.375 times less than his battle power, making it increase to match Goku's before Kaioken x4.

What follows now? He lost some power by transforming into Oozaru, which multiplied it by 10 to such amount that Kaioken x5 can't match.
EDIT: the follow up to these numbers. They were last standing with:

Goku - 4'000 -> 20'000 - the last number is Goku's hypothetical Kaioken x5, which sets the lowest that Vegeta can be as an Oozaru, if not higher.
Vegeta - 6'000 -> 4'000 - a sensible point to lower his power with the artificial moon, since Full Power Gohan (2'800) is still below him for when Vegeta reverts.

Oozaru - 40'000 -> 36'000 -> 3'600 - the only damage that Goku did was to his eye. Fortunately, this was kept when Vegeta reverted from Oozaru.
Gohan - 2'800 - I'm taking a shot at it, but this is Gohan with his power unleashed, which isn't considerably lower than Vegeta's, nor enough to beat him.

[apex_pretador]

so you want numbers with damage included (I always do include damage in my PL lists. See signature for full lists)

Goku :
8080 (Full power, pre battle)
KK = 12,120
KKx2 = 16,160

Vegeta :

18,000

______Goku decides to use 3x KK___________
Goku :
7,500
KK x3 = 22,500 (heavy strain)

Vegeta :

17,999

____________After vegeta is overpowered, he gets angry, and decides to burn earth__________

Goku :
20,000 (KK x3, dropping due to strain)

Vegeta - 17,400

____________Both charge beams_______________

Goku :
6,000
KK x3 - 18,000
KHH - 19,000 (couldn't powerup more due to strain)
KK x4 - 23,000

Vegeta - 17,400
Galick gun - 18,800

____________________vegeta hit_______________

Goku - 5,000

Vegeta - 10,000
____________________Vegeta decides to use moon ball____________________

Goku - 5000

Vegeta - 6000 , ozaru - 60,000
________________Vegeta loses tail_______________

Vegeta - 4,500

Goku - lol

_______________Vegeta hit by spirit bomb_____________

Vegeta - 2000

Gohan - 600 , ape - 6000
Yajirobe - 700
_________________vegeta cuts tail_________________

Vegeta - 1200

Vegeta (after) - 220
Krillin - 460

[Desassina]

so you want numbers with damage included (I always do include damage in my PL lists. See signature for full lists)

Goku :
8080 (Full power, pre battle)
KK = 12,120
KKx2 = 16,160

Vegeta :

18,000

Is this a Kaioken between 1 and 2, or just the initial boost before he's completely multiplied by KKx2?

______Goku decides to use 3x KK___________
Goku :
7,500
KK x3 = 22,500 (heavy strain)

Vegeta :

17,999

____________After vegeta is overpowered, he gets angry, and decides to burn earth__________

Goku :
20,000 (KK x3, dropping due to strain)

Vegeta - 17,400

This is the point that gets me the most, and the basis of my question and this thread: is Vegeta not that injured when Goku was able to graple with him, which includes gaining the upper hand? Was Goku still using KKx3, when the aura wasn't up around him? I like the rest of your numbers, but this point is crucial.

[apex_pretador]

so you want numbers with damage included (I always do include damage in my PL lists. See signature for full lists)

Goku :
8080 (Full power, pre battle)
KK = 12,120
KKx2 = 16,160

Vegeta :

18,000

Is this a Kaioken between 1 and 2, or just the initial boost before he's completely multiplied by KKx2?

I believe this is the basic kaioken (like what he used against nappa , or what king kai can do)
During his 6 day training till namek, he has mastered KK till x10 , so he doesn't need basic kaioken now.

______Goku decides to use 3x KK___________
Goku :
7,500
KK x3 = 22,500 (heavy strain)

Vegeta :

17,999

____________After vegeta is overpowered, he gets angry, and decides to burn earth__________

Goku :
20,000 (KK x3, dropping due to strain)

Vegeta - 17,400

This is the point that gets me the most, and the basis of my question and this thread: is Vegeta not that injured when Goku was able to graple with him, which includes gaining the upper hand? Was Goku still using KKx3, when the aura wasn't up around him? I like the rest of your numbers, but this point is crucial.

Goku's power was heavily dropping due to strain of KK x3. Vegeta didn't exchange blows for too long after being OP'd by goku. Had they continue to fight the same way, I think goku might've been in better condition. Had goku been able to handle kk x3 , he would've definitely got the upper hand. His body got bulky , didn't it? It is like side effect of heavy strain - freeza , grade forms etc.

[Darkprince410]

I believe this is the basic kaioken (like what he used against nappa , or what king kai can do)
During his 6 day training till namek, he has mastered KK till x10 , so he doesn't need basic kaioken now.

The base Kaiou-ken is the exact same as the 2x Kaiou-ken, as both are established to be a doubling of Goku's strength. Goku's later "use" of the Kaiou-ken 2x during the battle with Vegeta is contradictory to the manga narrator's statement of the strength boost the regular Kaiou-ken gives and what is shown later when Goku uses it in front of Ginyu.

Narrator: “Vegeta's strength has greatly surpassed what Goku had figured on...! That power was such that not even the Kaio-Ken, which doubled Goku's battle power, could give Goku the upper hand... What will you do?! Goku!!"

And this narrator statement is before Goku's erroneous use of the "Kaiou-ken 2x" so it's not an issue of getting the timing of the statement wrong.

[Desassina]

Hey, Darkprince, what do you think of the issue with Goku being able to grapple his way towards a head but, after he used KKx3 and wasn't surrounded by its aura? Still powered up?

[Darkprince410]

Hey, Darkprince, what do you think of the issue with Goku being able to grapple his way towards a head but, after he used KKx3 and wasn't surrounded by its aura? Still powered up?

I'm of the standpoint of, apart from when he's firing his Kamehameha during the Galick-ho/Kamehameha beam struggle, whenever he's not shown with the aura, he's not using the Kaiou-ken.