Ignore the above graph for now. I will explain how I got this.
Alright guys, I know you have seen several attempts at power scaling, but I promise you this one is different. We all know that official power levels stopped at the end of the Freeza saga in DBZ. Since then, there have been a lot of rumors and speculation circulating all over the internet mixed with opinions. Most people have given up on power scaling and say that the series is too inconsistent and in a mess. Today, I will attempt to see if we can use logic and mathematics to derive a scaling method that satisfies all information and feats from official sources, primarily from the manga, anime and Toriyama himself. Other sources will include the Daizenshuus, Databooks, Toei and other writers or authors. We will also use the anime and movies where needed but manga will always take precedence. There may also be some head canon. I don’t claim this to be the most accurate power scale, but just a different approach to power scaling that you may have not seen before that uses math to satisfy information we know from official sources. Hence, please keep an open mind!
Now, without further ado, let us begin!
After 17 years, in August 2014, the first DB movie was announced – Battle of the Gods (BoG), by Akira Toriyama himself! During that time, there was an interview with Toriyama:
Source: http://www.kanzenshuu.com/translations/ ... -toriyama/
Wait! In this interview, Toriyama gave us the first real indication of power scaling for Dragon Ball Super, stating that Goku is a 6, Beerus is a 10, and Whis is a 15. Now of course, this scaling is quite outdated and only represents where Goku was at during BoG. He’s definitely not a 6 in SSG anymore. However, we can still keep Beerus and Whis scales since they haven’t changed and try to derive where Goku is now. Most of you might be skeptic, but stick with me here till the end before you make up your mind.
If you treat these numbers as a linear scale, it will never work. For example if Goku used Kaioken, then 6 x 2 = 12 > Beerus (10). Thus, I will show you the logarithmic scale approach. Logarithms are used when the numbers become too big (like power levels). They are used today to measure earthquakes, pH, sound loudness (decibels), and light intensity.
As an example, the pH of water is 8 while the pH of vinegar is 2. This doesn't mean that vinegar is 8/2 = 4x more acidic than water. Actually, it is 10^8/10^2 = 10^6 or 1 million x more acidic!
Power scaling communities commonly use percentage differences based on official power levels. During the Freeza saga, SSJ Goku was 150 million while Freeza was 120 million. The percentage gap here is 25% i.e. SSJ Goku is 1.25 times stronger than Freeza during the Freeza Saga. Similarly, a difference of 0.1 on the log scale is 25%.
Just to be clear, a 1-point difference in a log scale means 10x stronger, a 0.1 difference is 1.26x stronger, 0.2 difference is 1.58x stronger, 0.3 difference is 2x stronger, a 0.4 difference is 2.5x stronger, a 0.5 difference is 3.16x ~ 3x stronger, a 0.6 difference is 3.98x ~4x stronger, a a 0.7 difference is 5x stronger, 0.8 increase is 6.3x ~6x stronger, 0.9 increase is 7.94x ~ 8x stronger.
Ok, we shall take the inverse log of each scale value:
Goku (SSG) = 10^6 --> 1,000,000 or 1 million
Beerus = 10^10 --> 10,000,000,000 or 10 billion
Whis = 10^15 --> 1 quadrillion
Remember these aren’t power levels. These are numbers converted using the inverse log scale. It is these numbers that we can use our multipliers before converting back to log scale. Note also that taking the log of a number less than 1 results in a negative number. This will happen for weaker characters or forms.
Super Saiyan God multiplier
Goku could not even touch Beerus as an SSJ3 and stated that even with fusion, he would have no chance against Beerus. Having fused before, he should have known the maximum potential of Vegito, which is a hypothetical SSJ3.
So SSG Goku > SSJ3 Vegito by logic.
In addition, the Daizenshuu 7, Base Vegito > SSJ3 Goku
I will assume that this statement refers to the point in time when Vegito made his appearance and not the Vegito at the end of DBZ. Note also that in the manga, Base Vegito never fought Buuhan. He immediately turned into a Super Saiyan. If Base was sufficient to defeat Buuhan, then there would no reason to turn into a Super Saiyan since it expends unnecessary energy and as Elder Kaioshin pointed out, can shorten the lifespan. Hence:
SSJ Vegito > Buuhan > Base Vegito > Buutenks > Ultimate Gohan > SSJ3 Gotenks > SSJ Gotenks ~ SSJ3 Goku (Buu Saga)
Base Vegito > SSJ3 Goku (Daizenshuu 7)
Base Vegito > Buutenks > Ultimate Gohan > SSJ3 Gotenks > SSJ Gotenks > SSJ3 Goku
Base Vegito > 1.5x Buutenks > 1.33x Ultimate Gohan > 1.25x SSJ3 Gotenks > 8x SSJ Gotenks ~ SSJ3 Goku
Base Vegito > 1.5x 1.33x 1.25x 8x SSJ3 Goku
Base Vegito > 19.95x SSJ3 Goku
Base Vegito ~ 20x SSJ3 Goku
SSG Goku (BoG)> SSJ3 Vegito (BoG)
SSG > 1.5x SSJ3 Vegito
SSG > 1.5x 400x 20x SSJ3 Goku
SSG > 1.5x 400x 20x 400x Base Goku
SSG > 4.8 million x Base Goku
SSG ~ 5 million (5E6) x Base Goku
This is where you'll disagree because 5 million is a very large number. It seems very large, but this is the only way to explain Goku's massive increase in power that eclipses any of his previous forms including fusion to stand a chance against Beerus, who is a universe buster level. Not only that, but it was stated both in the DBS anime and manga, that Base Goku (BoG) < Freeza (Namek). Hence, in order for that to be satisfied, a higher SSG multiplier is needed to jump from lower than Freeza to god level.
Using this, we can now reverse engineer the log scale for the other forms of Goku (BoG):
Base Goku = SSG / 5E6 = Log (10^6/(5 x 10^6) ) = -0.7
SSJ Goku = 50 x Base Goku = Log (50 x 10^-0.7) = 1.0
SSJ2 Goku = 2 x SSJ Goku = Log (2 x 10^1) = 1.3
SSJ3 Goku = 4 x SSJ2 Goku = Log (4 x 10^1.3) = 1.9
Remember that space scene where SSJ Goku fought Beerus and was actually doing better even without SSG? How can that be if he wasn't in SSG anymore? Piccolo sensed his ki so he couldn't have been using god ki right? Wait! I thought Goku absorbed the power of a SSG into his base. Shouldn’t his new base=SSG Goku (BoG)?
Toriyama stated in his interview that Goku absorbed the power of SSG. In the anime and movie, we see that SSJ Goku in space was able to spar with Beerus even without SSG. However, Piccolo could sense his ki. To prevent Earth's destruction, he dug deep into his body and added a fraction of the god ki he had as SSG into his SSJ form. Goku has only transformed into SSG for the first time but he has "felt" it but has no idea how to use it yet. It is not until after he trains with Whis that he learns to control this god ki and utilize 100% of that god ki into his SSJ form. This then creates SSB. Hence, the temporary SSJ form that Goku uses against Beerus in space is essentially an SSJ with some god ki or an incomplete SSB. This is my own headcannon to explain the anime scene. It's not needed for manga. Note that later in the ToP, Goku uses SSG as a separate form that is greater in power than Base, so that debunks Base=SSG.
To prove this theory, let's add 10% god ki to SSJ:
SSJ Goku (BoG + 10% God Ki) = Log (SSJ Goku (BoG) x10% SSG Goku (BoG) ) = Log (10^0.7 x 10% x 5E6) = 6.7 > SSG Goku (6)
We get the same value as SSG. To further validate this is if you change that 10% to 100%, it will be the same value as SSB.
Source: Episode 13, 19:30-19:50
Super Saiyan Blue Multiplier
SSB is the power of SSJ with SSG. So by logic, we'd multiply SSG by 50x to get SSB. These multipliers are official and were released in the Super Exciting Guide (SEG).
Source: Chapter 13, page 20
Here, Whis explains that because Vegeta used SSB against Cabba, he wasn't able to exert even 10% of the full power of SSB against Hit due to loss of stamina. It's due to this that Goku in SSG was even stronger than Vegeta was when in SSB when facing Hit. Note very carefully though that Whis says Vegeta is not even 10% his full strength. If we are to say that SSB = 50x SSG, then SSG > 2% SSB or > .002x SSB. Since 2% is smaller than 10%, this is possible and doesn't contradict what Whis says.
Let's say that from BoG to RoF, Goku got 25% (1.25x) stronger after training with Whis.
So, SSB Goku (RoF) = 50 x SSG x 1.25 = Log [50 x 10^6 x 1.25 ]= 7.8
Golden Freeza >= 1.25x SSB Goku (RoF) = 7.9
Now back again to the "Saiyan Beyond God", both in the DBS anime and the RoF manga spin-off, Base Goku fought Final Form Freeza. In RoF, prior to meeting Freeza, Goku and Vegeta were both training in a dimension created by Whis's staff. Here, it was difficult for them to move unless they controlled their ki. It's here for the first time Vegeta felt god ki and utilized it without having to undergo the SSG ritual. Goku also learned to master his god ki and it was shortly after this, that he fought Freeza immediately. Thus, he already has using god ki in his base. It was thought again that this is a "new" Base Goku with the power of an SSG but really it's just god ki. Whether or not you want to buy into the "two base theory", I shall use the exact same headcanon above and add 10% god ki into Base, but since this time it's from Base and not SSJ, we multiply instead of add:
Base Goku (Base (RoF) x10% god ki) = Log (Base Goku (BoG) x 1.25 x 10% x 5E6) = Log (10^-0.7 x 1.25 x 10% x 5E6) = 5.1
Again, to validate, the 10% god ki can be changed to 100% and it should be SSG, which it is (6.1).
After RoF, Goku trained in the ROSAT for 3 years! Then, he fought Hit, Black, and Jiren. Did he get zenkais for all? The truth is that Goku has become too strong and after a point, you reach saturation point for zenkais. This means it becomes more difficult to get a zenkai the stronger one gets, which makes sense (see graph below). For example, in Namek, Goku went from 90,000 to 3 million in his base alone due to zenkai boost. He can still get stronger through traditional training.
In Chapter 20 of Dragon Ball Super, Trunks states that Goku and Vegeta have both pushed their bodies to their limits, so it’s not that easy to get zenkai boosts anymore.
Yet Black was able to still receive zenkai boosts while in Goku’s body. Why? Black himself stated in Chapter 20 that when he took Goku’s body, he was not able to access all of its power. He wasn’t even able to go SSJ. This is nothing new. Ginyu had the same problem back in the Freeza saga. Apparently, Black was able to discover a shortcut to quickly gaining access to the full power of Goku’s body – zenkai boosts. Each time Black healed himself from the brink of death, the healing regenerated new cells that used Zamasu’s own ki. This allowed Zamasu to make Goku’s body more his own. It’s actually the reason why he changed into Super Saiyan Rosé instead of Blue. In the anime, Black got stronger after taking damage but zenkai doesn’t work like that.
Source: Chapter 20, DBS Manga
Now that we understand how difficult it is to get stronger, let's say Goku got 15% stronger each year for 3 years so 1.10^3=1.33x. Then after fighting Hit, he got 1.25x stronger. Then after fighting Zamasu / Black, he got 1.33x stronger and finally, after the Tournament of Power, he got 1.33x stronger. This is 1.25x from BoG to RoF, 1.33x from RoF to U6 arc, 1.25x from U6 arc to Zamasu arc, 1.33x from Zamasu to ToP, and 1.33x Post-ToP. So total is 1.25x 1.33x 1.25x 1.33x 1.33x = 3.68x ~4x from BoG to Post-ToP. Doesn't seem like a lot does it? Again, remember that there aren't any free zenkais anymore.
Base Goku = Base Goku (BoG) x 1.25x 1.33x 1.25 = -0.4
SSB Goku = Base Goku x 50 x 5E6 = 8.0
CSSB Goku = SSB Goku x 20 = 9.3
The Completed SSB (CSSB) is a manga exclusive transformation that doesn't appear in the anime. Rather, in the anime, we see Goku used Kaioken x20 and Vegeta uses the SSBE. In fact, in Chapter 35 in the manga, Goku used CSSB against Jiren analogous to Episode 109 of the anime where he used SSB+Kaioken x20. The exact same dialogue between Krillin and Whis is used between the two, so it can be assumed here that CSSB is indeed equivalent to the anime’s version of SSB+Kaioken x20.
In addition, the director of Episode 131, Megumi Ishitani, stated herself that the SSBE is equivalent to Goku's SSB Kaioken x20:
While it's not necessary that a 20x multiplier is also the case for CSSB, I will use it anyways in this guide.
Ok, let’s do Potara Fusion now. We have all heard what Vados said in Episode 114,
the fusion creates a new body and personality greater than the sum of the two individuals and increases their power by tens of times.
Now before we jump the gun, let’s really analyze this. What is meant by “tens” of times? Tens is lower than “hundreds” of times. Tens is also plural so it does not mean 10. It means 20, 30, 40…. up to 90 but not 100. Therefore, the maximum multiplier can be 90. If we take Vados’s words literally, it means:
Potara Fusion = (A + B) x C where C = 10-90.
Let me tell you why this doesn’t make sense. In the Buu saga, we were first introduced to Potara fusion. Vegito in his base form was much stronger than SSJ3 Goku (Daizenshuu 7). If that is true, then
Base Vegito > SSJ3 Goku
(Goku + Vegeta)x C > 400x Goku
(Goku + Goku)x C > 400x Goku
2x Goku xC > 400x Goku
C > 200
What does this mean? It means that in order for Vegito in his base to be stronger than SSJ3 Goku, C has to be a value greater than 200! If we went by Vados, C can only be 10-90!
Furthermore, before I forget, in the SEG, it stated that the Fusion formula was AxB which also does not make sense. Multiplying together is essentially squaring Goku’s base power level which would give an enormous power rivaling even Whis. In statistics, the cross doesn't have to mean multiplication but can also mean "by" which relates two variables. https://en.wikipedia.org/wiki/Multiplication_sign
Now let's go back to the originally derived relationship between Base Vegito and SSJ3 Goku:
Base Vegito = 20x SSJ3 Goku
(A+B)xC = 20x 400x Base Goku
(Goku+Goku) x C = 8000x Goku
2xC = 8000
C = 4,000
Potara Fusion = (A+B) x C where C=4,000.
Base Vegito (Zamasu arc) = (Base Goku + Base Vegeta) x 4,000 = Log [(10^-0.4 + 10^-0.4) x 4,000] = 3.5
SSB Vegito = Base Vegito x 5E6 x 50 = Log [10^3.6 x 5E6 x 50] = 11.9 > Beerus = 10
This makes sense because in the manga, Chapter 23, page 43, Shin states that SSB Vegito surpassed Beerus.
Whether or not you believe Shin, he is a Shinjin that can sense god ki and has lived for more than 5 million years.
SSJ Vegito = 50 x Base Vegito = Log [50 x 10^3.6] = 5.2
SSJ2 Vegito = 2 x SSJ Vegito = Log [ 2 x 10^5.2] = 5.5
SSJ3 Vegito = 4 x SSJ2 Vegito = Log [ 4 x 10^5.5] = 6.1
And this validates that SSG Goku (6.3) > SSJ3 Vegito (6.1)!
Another fusion we can calculate really easy now is Zamasu! If we take SSR Black = 1.5x SSB Goku (Zamasu), then SSR Black = 8.2. Then, if we take Zamasu = SSJ2 Goku (Zamasu arc) / 1.5 then Zamasu = 1.4
For Vegito, we used C=4,000. The multiplier will be different for different people. We know that the closer the physical characteristics and battle power between the fusees, the stronger the fusion. Toriyama himself confirmed that similar size and power is all that is needed. Factors like "personality" and "rival boost" were made-up things by fans who misinterpreted Elder Kai's words in the original DBZ manga. In fact, what the Elder Kai meant was that Goku and Vegeta live to outdo each other which helps them sustain similar battle powers.
Zamasu and Black (Goku's body) are completely different. Zamasu is a Shinjin and Goku is a Saiyan. They have drastically different battle powers too. The difference between Zamasu and Black is the difference between a SSJ2 and SS Rose. Therefore, both battle powers and physical characteristics are different. They also fused together with Black already in SS Rose. Elder Kaioshin stated not to fuse as Super Saiyans as that shortens the lifespan:
Source: Ch. 503
In the DBZ Manga, Goku himself thought that fusing with someone significantly weaker like Mr. Satan would make the resulting fusion weaker:
Hence, it wasn't a perfect fusion. Yes, technically Zamasu and Black are the same person soul wise, but that doesn't matter. The power gap is immense and hence, the multiplier will be significantly lower. In addition, in the manga, MSSB Goku fought Merged Zamasu evenly.
So we'll use a C=10 which is 1/400th of the full power of Potara.
Merged Zamasu = (Zamasu+SSR Black) x10
Merged Zamasu = log [(10^1.5 + 10^8.2) x10] = 9.2 = CSSB Goku (Zamasu arc)
Since we are on the subject of Zamasu, might as well do Future Trunks! We have clearly seen SSJ2 Future Trunks in Chapter 15 of the manga, spar equally with SSJ3 Goku. The SS Rage form of Trunks will not be used as it is anime exclusive.
SSJ2 Future Trunks (Post-Zamasu) ~ 2x SSJ3 Goku (Zamasu)= 2.5
-----------------------------------------Tournament of Power------------------------------------------------------------------------------
Base Goku (Post-ToP) = Base Goku (Zamasu arc) x1.33 x1.33 = Log (10^-0.4 x1.33 x1.33) = -0.2
SSJ Goku = Base Goku x 50 = 1.5
SSJ2 Goku = SSJ Goku x 2 = 1.8
SSJ3 Goku = SSJ2 Goku x 4 = 2.4
SSG Goku = Base Goku x 5E6 = 6.5
SSB Goku = SSG Goku x 50 = 8.2
CSSB Goku = SSB Goku x 20 = 9.5 = SSB Kaioken x20 = SSBE < Beerus (10)
As for Jiren, Whis said that he is the mortal that surpasses a Hakaishin out of all 12 universes, even the four that were excluded. I will hence assume: Jiren > SSB Vegito (12.1) > Beerus (10)
To simplify, Jiren (ToP) = 13
It might seem like a lot but given what Jiren is capable of - even going head-to-toe with mastered UI Goku, I don't think it's too wild at all. The next on the list is Ultra Instinct Goku. The fight between MUI Goku and Jiren was not equal but it wasn't completely one-sided either. We shall give MUI Goku = 2x Jiren.
Ultra Instinct Goku (ToP) = 13.2 (1.5x Jiren)
Ultra Instinct Goku (Post-ToP) = UI Goku (ToP) x 2 = 13.3
Next up, Android 17. In Chapter 31 of the DBS manga, Android 17 was shown to be sparring with SSJ3 Goku evenly. Much different than the anime that portrayed SSB Goku vs 17. The scaling makes much more sense in the manga. Hence, we shall assign Android 17 the same log scale as SSJ3 Goku.
Android 17 = 2.5 =~ SSJ3 Goku (Post-Zamasu)
Furthermore, Toshio commented on his twitter that Gohan = 17>Frost>Piccolo.
Frost=1.4 (1.5x weaker than SSJ Goku (ToP))
Piccolo=1.2 (1.5x weaker than Frost)
Kefla is next, but to get her, we have to calculate Kale. According to Chapter 37-38 DBS Manga, Golden Freeza said that her strikes hurt more than SSJ Caulifla's, confirming that Base Kale > SSJ Caulifla. When she went BSSJ, she easily overpowered Golden Freeza and SSB Goku. Since Kale is a Saiyan, she would also receive a zenkai boost. Since Kale is a Saiyan and she was pushed to her limits and her zenkai isn't saturated like Goku's, I'll give her a zenkai of 2x post-ToP.
Hence, if we take Base Kale > 2x SSJ Caulifla and Base Caulifla = Base Goku (ToP) /1.5, then:
Base Caulifla = Base Goku (ToP) /1.5 x2 (zenkai) = -0.2
SSJ Caulifla = Base Caulifla x 50 = 1.5
Then Base Kale (Post-ToP) = log ( 10^1.5 x2 x2) = 1.8
SSJ Kale = log (10^1.8 x 50) = 3.5
If BSSJ Kale > 2x SSB Goku, then BSSJ Kale = 8.5 > SSB Goku (8.2)
To reverse engineer, we can calculate the BSSJ multiplier:
BSSJ multiplier = BSSJ Kale / SSJ Kale = 10^8.5/10^3.5 = 10^5 = 100,000.
The same rules for Potara Fusion apply that applied before. Kale and Caulifla have the same physical characteristics but there is a large difference in battle power. They fused while transformed. Thus, this is the difference between SSJ Caulifla and BSSJ Kale. Base Kale was already stronger than SSJ Caulifla, so this difference is enormous. Due to this, they can't receive the full 4000. I'll give her 20 which is 1/200th of the full power. This makes it consistent with Vados' "tens" statement and also puts both her and Gohan below Beerus.
Then BSSJ Kefla = (SSJ Caulifla + BSSJ Kale) x 20 = Log [ (10^1.5 + 10^8.5) x20] = 9.8
Note that there's no Base Kefla because if the fusees are already Super Saiyan, then the resulting fusion is also Super Saiyan.
Since Gohan and Kefla fought equally and did a double K.O.:
Ultimate Gohan (Post-ToP) = 9.8
Hakaishin Toppo = 9.4 = SSBE Vegeta (ToP) = CSSB Goku (ToP) < CSSB Goku (Post-ToP)
Kale is very similar to Broly. In the trailer, we see that Base Broly with his collar overpowers SSJ Vegeta, already stating that Base Broly with collar > SSJ Vegeta. Later, Base Broly without his collar is seen fighting SSB Goku equally in a false SSJ state. This implies that Base Broly without collar > SSG Goku. Then Broly transforms further into Super Saiyan Full Power. Let's say Base Broly (with collar)> 2.5x SSJ Goku (Post-ToP). Also the false SSJ state will lie somewhere between his SSJ and Base without collar. Both Kale and Broly also have a regular Super Saiyan form as an intermediate.
Base Broly (with collar) = SSJ Goku (Post-ToP) x2.5 = 1.9 > SSJ Goku (2.0)
Base Broly (without collar) = 50,000 x Base Broly (with collar) = 6.6
False SSJ Broly = Base Broly (without collar) x 75% x 50 = 8.2 = SSB Goku (Post-ToP)
SSJ Broly = log(10^6.6 x 50) = 8.3 > SSB Goku (Post-ToP)
BSSJ Broly = log(10^8.3 x BSSJ Multiplier derived from Kale ~ 100,000) = 13.3 = Ultra Instinct Goku (13.3) > Jiren (13) > Beerus (10)
The 50,000 factor used for the collar was chosen such that Base Broly (without collar) = SSB Goku.
Note that in Episode 114 of the anime, a third form of Kale's was shown in which she was able to control her Berserker state. In this form, she became slimmer. Krillin then goes on to state that she's not Berserk anymore and that she has this power under control. Tenshinhan even says that her energy has massively increased. While this may imply that this third, slimmer form, is stronger than her Berserker form, this isn't true. Tenshinhan's words compared her to the previous SSJ state. In Dokkan Battle, this form is actually called her "Super Saiyan 2" form. If this is true, it would make sense since in both the anime and manga, BSSJ Kale was able to overpower SSB Goku while SSJ2 Kale was able to be defeated by SSG Goku easily. In the Blu-Ray Box 7 of the DBS Anime, there's a "Setting Art Book" in which character sheets of Kale are shown that show her base, SSJ, and Berserker.
Source: DBS Setting Artbook, Blu-Ray Volume 7
Since we have derived Potara Fusion, let us derive Fusion Dance!
Before I start, I should point out that Gogeta is not canon and was a character introduced by Toei in Movie 12. Gogeta re-appeared in GT, also created by Toei. Hence, to scale Gogeta, we are allowed to use sources by Toei, since he was literally created by them.
Here are the GT Perfect Files:
The above is from GT Perfect Files Volume 2. The green box above translates to:
“This is the fusion of Super Saiyan 4 Goku and Super Saiyan 4 Vegeta. The strongest warrior in the entire Universe, and probably in every dimension!! It's Fusion of unprecedented power that lasts 30 minutes, however, it's only weakness is that lasts 10 minutes due to the Super Saiyan 4 state. His power is perhaps many tens of times stronger than a single Super Saiyan 4.”
Again, in GT Perfect Files Volume 2:
Again, the “tens” of times is stated which was said by Vados before. It’s ambiguous but “decina” means ten in Italian and that is clear. This means that SSJ4 Gogeta=tensxSSJ4 Goku. If this is true, then the we can derive the formula for Fusion Dance as follows:
SSJ4 Gogeta = (A+B)xC = (tens)xSSJ4 Goku
If A and B represent SSJ4 Goku and SSJ4 Vegeta where A~B, then:
SSJ4 Gogeta = (SSJ4 Goku + SSJ4 Vegeta) x C where C=10-90
SSJ4 Gogeta = (SSJ4 Goku + SSJ4 Goku) x (10-90)
SSJ4 Gogeta = 2 x SSJ4 Goku x (10-90)
SSJ4 Gogeta = (20-180) x SSJ4 Goku
As discussed above, this range of C depends on the accuracy of the Fusion Dance as well as how similar the battle power and physical characteristics are! If you perform the dance wrong, then C can be between 0-1, which makes their power LOWER than the sum of their individual battle strength. If you touch with a fist and finger, you get a Fat version while if you don't touch with the fingers, you get an old skinny version of the Fusion that is weaker than the fat one. Doing the dance correctly, but not having similar battle power or physical characteristics, can make C=10-90. Goku and Vegeta, while both Saiyans with similar battle power, still have differences. A perfect fusion would be one where the fusees are twins which would give a C=90. In this case, we can give a C=50 for Gogeta. A C=50 would make Gogeta = 100xGoku/Vegeta. Gogeta also states 100x Big Bang Kamehameha so this make sense!
So, let’s do this:
Base Gogeta (Post-ToP) = (Base Goku + Base Vegeta) x50 = Log [ 100 x 10^-0.2 ] = 1.8
SSJ Gogeta = Base Gogeta x 50 = 3.5
SSJ2 Gogeta = Base Gogeta x 2 = 3.8
SSJ3 Gogeta = Base Gogeta x 4 = 4.4
SSG Gogeta = Base Gogeta x 4E7 = 8.5
SSB Gogeta = SSB Gogeta x 10 = 10.2
CSSB Gogeta = SSB Gogeta x 20 = 11.5 ~ SSB Gogeta + Kaioken x20 < BSSJ Broly (13.3)
SSJ Gogeta = 100x Base x50 = 5000x Base = 12.5x SSJ3 Goku
--------------------------------------------------DRAGON BALL GT---------------------------------------------------------------------------------------
Since we are on the subject of GT, let's do SSJ4. The SSJ4 is a transformation from Golden Oozaru not from SSJ3. The Golden Oozaru is implied to be an Oozaru that has just turned Super Saiyan, and hence the fur has become yellow. Thus,
Super Oozaru = 50x Oozaru = 500x Base
Super Oozaru 2 = 2x Super Oozaru = 1000x Base
Super Oozaru 3 = 4x Super Oozaru 2 = 4000x Base
So "Golden Oozaru" is not a fixed multiplier and is based on what state of SSJ is being used while in Golden Oozaru. The only way to determine this is by the hair but since the hair doesn't change much between SSJ1 and SSJ2, it can't be distinguished that easily.
According to the GT Perfect Files, Golden Oozaru Baby Vegeta > SSJ4 Goku. It is not until Goku absorbs the energies of Gohan, Goten, Trunks, and Pan that he is able to overpower Super Oozaru Baby Vegeta. Since Oozaru is a 10x multiplier, it means that the gap between SSJ4 Goku and Super Baby cannot have been more than 10x. The question is then how strong was Golden Oozaru Goku? Was he a regular SSJ, SSJ2, or SSJ3 in Golden Oozaru? Several people have theorized that Goku was a SSJ3, which would make the multiplier 400x10=4000x. However, based on the recent design of Golden Oozaru Cumber and previous designs of Golden Oozaru including Baby, Broly, and Gogeta, one thing that is consistent is the hair. The hair remains the same even in Oozaru. Hence, Goku could not be in SSJ3 because he doesn't have the long hair while in Oozaru. However, as the fight between Golden Oozaru Goku and Super Baby progressed, Goku's ki continues to rise. The only way to make sense of this is that, Goku did surpass SSJ2 while in Golden Oozaru and approached but did not reach SSJ3 as he could not control his power.
Adult Goku (53 years old) ~ Kid Goku (12 years old) - GT Perfect Files state that they are equal in power except stamina issues.
Adult Goku ~ 2x Vegeta
Baby Vegeta = 16x Vegeta ~ 8x Kid Goku
SSJ Baby Vegeta = 50x Baby Vegeta ~ 800x Vegeta ~ SSJ3 Kid Goku ~ 400x Kid Goku
Super Baby (Strongest Form 1) ~ 2x SSJ Baby Vegeta (After absorbing energies of Gohan, Goten, Trunks, and Bulla)
Super Baby (Strongest Form 2) ~ 4x Strongest Form 1 (After absorbing the energies of all people on Earth) ~ 3200x Kid Goku < Golden Oozaru Goku < SSJ4 Goku = 4000x Kid Goku
Golden Oozaru Goku = 500x to 4000x Kid Goku
Golden Oozaru Baby Vegeta = 10x Strongest Form 2 ~ 4000x Baby Vegeta ~ 32,000x Kid Goku > SSJ4 Goku ~ 4000x Kid Goku
Golden Oozaru Baby Vegeta ~ 8000x Vegeta ~ 32,000x Kid Goku < Full-Powered SSJ4 Goku ~ 40,000x Kid Goku
Hence, to make sense of this, I will say that:
SSJ4 = Condensed Golden Oozaru = 8x Super Oozaru = 4000x Base (Implied that Goku added the power of SSJ3 into Oozaru.)
Full-Powered SSJ4 Goku = 10x SSJ4 = 40,000x Kid Goku (After absorbing energies of Gohan, Goten, Trunks, and Pan)
The reason that Goku got a 10x multiplier when absorbing Gohan, Goten, Trunks, and Pan's energies versus Baby Vegeta who only got a 2x multiplier when absorbing the energies of Gohan, Goten, Trunks, and Bulla's energies is because when Baby Vegeta received their energies, none of them were Super Saiyan. This may be due to the fact that Gohan, Goten, and Trunks were already depleted of their energies after fighting with Goku and Uub and/or that their full power could not be drawn out without Baby inside their bodies. The egg inside their minds was not enough.
By the way, according to herms98 translation of the DBGT anime comics, a scan revealed that Base Vegito > SSJ4. This source is not stating that Base Vegito (Buu Saga) > SSJ4 Goku (GT) but rather a general statement that the current fusion of Goku and Vegeta still would perhaps surpass the SSJ4 form. (Source: https://twitter.com/Herms98/status/675785758473490432) This is consistent with what we calculated since Base Vegito ~ 20x SSJ3 Goku which is 2x SSJ4.
Once GT starts, Kid Goku was shown to fight Rilldo who was stated to be stronger than Kid Buu. Since SSJ3 Goku (Buu Saga) ~ Kid Buu, then Base Kid Goku > Rilldo > SSJ3 Goku (Buu Saga). Four years pass after the defeat of Buu to the start of BoG. Six years later is the 28th Tenkaichi Budokai where Goku meets Uub. Then Goku trains 5 more years with Uub in the ROSAT.
Since Goku (GT) > Rilldo > Kid Buu ~ SSJ3 Goku (Buu Saga)
Then Goku (GT) > 1.3x Rilldo > 1.25x Kid Buu ~ SSJ3 Goku (Buu Saga)
Goku (GT) > 1.3x 1.25x 400x Goku (Buu Saga)
Goku (GT) > 650x Goku (Buu Saga)
If Goku got 10% stronger for 4 years from Age 774 to Age 778, then 70% stronger for 6 years from Age 778 to Age 784, and finally 80% stronger for 5 years training with Uub from Age 784 to Age 789, then 1.1^4 x 1.7^6 x 1.8^5 =~ 667x > 650x.
Now let's say that after Goku's battle with Baby Vegeta, Super 17, and Evil Shenrons, he received a 10% (1.1x) boost from each one, so 1.1^3. Note that not a lot of time passed between these villains and Goku's zenkai by the same logic is also saturated. He can only get stronger through good old training. Goku: Xeno is a Goku summoned by Chronoa in Age 850 after she looks through Future Trunks memories. DBGT ends in Age 790 with Kid Goku leaving with Shenron. We don't see him until a 100 years later. However, since 850-790 = 60 years, Chronoa may have summoned Goku 60 years after Age 790 to the Time Nest. Remember that Goku was turned into a kid so he could still have aged 60 years without looking old. Then, we can give him a 10% increase each year for the remaining 60 years, so 1.1^60 to get Goku: Xeno.
Kid Goku (GT) = Log [1.1^4 x 1.7^6 x 1.8^5 x 1.1^3 x Base Goku (Buu Saga) = Log (1.1^4 x 1.7^6 x 1.8^5 x 1.1^3 x 10^(-0.9) ) = 2.1
Goku: Xeno (60 years after GT) = Log (Kid Goku (GT) x 1.1^60) = 4.6
SSJ4 Goku: Xeno = 4000x Goku: Xeno = Log (4000x 10^4.6) = 8.2 = SSB Goku Post-ToP (827)
Well, that's it!
My raw data and calculations: https://www.dropbox.com/s/1wr2ll16h4rss ... .xlsx?dl=0
This is a graph of characters we know and have been revealed in Dragon Ball Super. I didn’t put characters like Gogeta since it’s not canon, even though Gogeta can be scaled. The anime does a lot of fan service and hence, until the manga catches up to the anime, characters like Kefla and Kale are based on educated guesses. Characters like Android 18 who hasn’t gotten any stronger since the Cell saga are too weak based on this log scale and would be below 1 and may even be negative.
Lastly, the point of this was to use math to see if there was a solution that meets all conditions. Sometimes there is no solution and that's when you have a plot hole.
----------------------------------PART 2 BEGINS---------------------------------------------------------
Super Saiyan Grades, Multipliers and Completed SSB
Since Super Saiyan was first achieved, we have seen different grades of Super Saiyan during the Cell saga. This includes grade 2 or Ascended SSJ (ASSJ) by Vegeta after he first came out of the Room of Spirit and Time with Trunks to fight Cell. Next was grade 3 or Ultra SSJ (USSJ), the form Trunks used which greatly increased his power at the expense of speed. In the DBS anime, Trunks even learned to trick Vegeta by using this form to bait him and then switch back to SSJ2. The Super Saiyan Rage form he used against Zamasu seemed to be a combination of grade 3 and SSJ2. Last was grade 4 or Fully powered SSJ (FPSSJ) when Goku and Gohan were able to maintain a SSJ for an entire day doing normal activities like taking a bath. Goku released his full power against Perfect Cell, showing to everyone how much stronger he has gotten. These grades are not fan-made but were written in an official source for the Trunks anime comics translated by Herms: viewtopic.php?t=22411
Oozaru = 10x Base
Kaioken x N = Nx Base where N=2-20 (N>20 is possible but never shown)
SSJ Grade 1 = 50x Base (power, speed, stamina balanced but slight agitated state and stamina drain for all newly transformed SSJs)
SSJ Grade 2 (ASSJ) = 60x Base (Slight increase in power and speed, abandoned due to high energy consumption and stamina drain)
SSJ Grade 3 (USSJ) = 10x SSJ (Lots of power, Increased muscle mass lowers speed by a lot, very high energy consumption and stamina drain)
SSJ Grade 4 (FPSSJ) = 50x Base (power, speed, and stamina perfectly balanced)
Alternatively, some people have scaled SSJ Grade 3 = 75x Base if you don't agree with "Dragon Ball Legend of Manga" released in 2007 by Shueisha with original language Japanese which states 10x SSJ. It was translated to Spanish and French.
Source: viewtopic.php?t=38328 https://lh3.googleusercontent.com/liNSB ... h768-rw-no
SSJ2 (SSJ Grade 5) = 2x SSJ
SSJ2 Grade 2 = 4x SSJ
SSJ2 Grade 3 = 8x SSJ ~ SSJ3 (Future Trunks and Vegeta have both shown to surpass SSJ3)
SSJ2 Grade 4 (SS Rage) = 10x SSJ2 (anime only, higher grade of SSJ2, exclusive to Future Trunks.)
SSJ3 = 4x SSJ2 (Lots of power but lots of stamina drain)
Fusion Dance = (A+B) x C where C = 10-90 based on similarities between fusees (physical characteristics and battle power)
C = 0 - 1 when dance is performed incorrectly.
Potara Fusion = (A+B) x C where C = 1-20,000 based on similarities between fusees (physical characteristics and battle power)
Ex. Fusing with Farmer with shotgun with battle power = 5, will not automatically give C > 1 for free when the difference in power is more than millions. The larger the difference between the fusees, the lower the C.
SSG (Base + 100% God ki) = 5,000,000x Base (Derived above)
SSB (SSJ + 100% God ki) = 50x SSG (Derived above)
CSSB = 20x SSB (AKA. Completed Super Saiyan Blue. Equivalent to Kaioken x20 and SSBE in the anime)
https://twitter.com/Herms98/status/1028025720751017984 - Quote from the director of Episode 131, Megumi Ishitani confirming it is equal.
SSJ (+10% God Ki) = SSG (What Goku used against Beerus in space. Anime only.)
Base (+10% God Ki) = SSG/10 (AKA Saiyan Beyond God. What Goku uses against Freeza in RoF.)
Ultra Instinct = ? x Base (May not even be a power-up in the manga but a technique that allows one to move instinctively and land critical blows. However, the "heat" accumulated is the result of gathering the kinetic energy around oneself from moving instinctively.)
BSSJ (Berserk SSJ) = 100,000x SSJ (Similar to USSJ, but no speed drop and more power than SSJ3. Multiplier is still less than SSG. Only applies to Legendary Saiyans like Kale and Broly.)
Super Oozaru = 50x Oozaru = 500x Base
SSJ4 = 8x Super Oozaru = 4000x Base (Condensed Golden Oozaru, 10x SSJ3)
Full-Powered SSJ4 = 10x SSJ4 (After absorbing energies of SSJ Goten, SSJ Trunks, Pan, and SSJ Gohan)
We've seen different grades of SSJ2 as well. We saw that Future Trunks in SSJ2 was able to fight on par with SSJ3 Goku and saw that Vegeta in SSJ2 after Bulma got slapped by Beerus was greater than SSJ3 Goku. This means that it is possible for characters to gain mastery over a transformation to reach higher grades. Characters like Vegeta and Trunks who never achieved SSJ3, trained more in their SSJ2 form and hence, is at a higher grade. The goal of any form is to have maximal power output without stamina drain.
In addition, in Super, we did not see ASSJ or FPSSJ from Caulifla or Cabba with Caulifa skipping directly to SSJ2. USSJ still exists as a distinct transformation however since we have seen both Future Trunks and Caulifla perform it. In addition, FPSSJ is a combination of good ki control and maintaining the SSJ state longer to consume less stamina. It is akin to the Completed SSB state or CSSB.
Read here for the theory on ultra instinct: viewtopic.php?f=8&t=40707
Fans always like to overestimate and exaggerate destructive capacity. This term has been used by Marvel, DC, and other fictional verses to gauge power of characters. The primary issue with this form of measurement has always been that our own universe is vastly different from the fictional verses. Thus, we can't easily use the same number of stars, galaxies, solar systems in our universe to equate that of another. In Dragon Ball, things like the Earth, Moon, and Sun however are consistent so the mass of each can be used.
In Dragon Ball, Jackie Chun aka Master Roshi, in the 21st Tenkaichi Budokai (Chapter 52), was able to destroy the moon with a Super Kamehameha. According to the V-Jump and the Daizenshuu, Master Roshi’s battle power at that time was 139. If you're having difficulty accepting these as sources, remember that V-Jump is the very magazine that the Dragon Ball manga was published in each week. The Earth has a mass (5.972 x 10^24 kg) approximately 81 times larger than the moon’s mass (7.347 x 10^22 kg). Therefore, 81x139 = 11,259 ~ 11,000. This would be the minimum battle power necessary to destroy the Earth. Why am I using mass instead of volume or density? Because mass is directly proportional to energy via E=mc^2 where c=speed of light. If I had used density, the Earth's density (5.51 g/cm^3) is only 1.65 times the Moon's density (3.34 g/cm^3). Obviously if Roshi is only 1.65 times stronger, he can't destroy the Earth. If I used volume, 50 Moons can make up Earth. Fifty is not a bad number to work with but as I said earlier, mass has a better relationship with energy than volume. During the Saiyan Saga, Vegeta with a battle power of 18,000 claimed he could destroy the Earth with his Galick Gun. In addition, we saw Freeza in his first form with a battle power of 530,000 easily destroy Planet Vegeta which is assumed to be the same size of the Earth.
The Sun has a mass of 1.989 x 10^30 kg which is approximately 333,054.243 times larger than the mass of the Earth. If 11,000 BP is the minimum needed to destroy Earth, then 333,000 x 11,000 =~ 3.67 billion BP is necessary to destroy a star the size of the Sun. Freeza’s first form (530,000 BP) thus would have a destructive capacity of 530,000/11,000 =~ 48 Earth-sized planets while Vegeta (18,000 BP) would have a destructive capacity of 18,000/11,000 =~ 1.6 Earth-sized planets. Freeza’s fourth form (120 million BP) is 120 million / 11,000 =~ 10,909 Earth-sized planets which is still lower than the destructive capacity of the Sun. Freeza final form would have to be at least 33x stronger to destroy the Sun.
Super Perfect Cell claimed that he is capable of destroying the solar system with his Kamehameha. A solar system comprises of the star with the orbiting planets. In the case for our solar system, that’s the Sun plus 8 planets. However, just the sheer destructive force of the Sun exploding would also destroy the orbiting planets. We can conclude that Cell's Kamehameha would have simply hit the Sun, destroying it and the solar system. Therefore, the destructive capacity for a solar system will still be equivalent to a single star and it would require Perfect Cell to be at least 333,054/10,909 =~ 33 times stronger than Freeza’s final form to be able to do this. This seems reasonable given that Super Perfect Cell > Perfect Cell > Semi-Perfect Cell > Imperfect Cell ~ Android 16 > Android 17 > Android 18 > SSJ Vegeta > Freeza. To put some numbers on this, Super Perfect Cell > 2x Perfect Cell, Perfect Cell > 2x Semi-Perfect Cell, Semi-Perfect Cell > 2x Imperfect Cell, Imperfect Cell > 1.5x Android 17, Android 17 > 1.25x Andriod 18, Android 18 > 1.5x SSJ Vegeta, SSJ Vegeta > 1.5x Freeza. Thus, 2x 2x 2x 1.5x 1.25x 1.5x 1.5x = 33.75x which is what we calculated earlier!
Many fans put Kid Buu as a galaxy buster (I used to as well), but nowhere in the DBZ manga or any guidebook has it been stated that he has accomplished this feat in the past. The anime always puts extra information or scenes not found in the manga. The manga is by the original author and it should take precedence. There's a lot of misinformation on the net, so go to the original source always and use Viz translations.
In Episode 9 of Dragon Ball Z Kai and Chapter 446 of the DBZ Manga, it is stated by Shin that Kid Buu destroyed "hundreds of planets" in a few years. In the anime however, we see a blue galaxy in the background disappear. This is the source of the confusion.
That blue galaxy in the background was added into the anime being erased because gradually, more planets were being wiped out in that galaxy. The less space debris and things orbiting the center black hole of a galaxy, the more it's going to look like it fades away over time. The entire galaxy wasn't wiped out by a single attack.
The final fight between SSJ3 Goku and Kid Buu took place on the Kaioshin realm. Here, in Chapter 510 of the DBZ manga, the Elder Kaioshin tells Goku to do what he wants and that “this world won’t break so easily!”.
The Kaioshin realm is 1/10th the size of Universe 7 and orbits the universe as stated in the Daizenshuu 4.
As far as what occurred, the Kaioshin realm withstood their fight and did not get destroyed. In Dragon Ball Super episode 31, Jaco explains that the universe contains a tremendous number of galaxies, and that his Galactic Patrol unit only covers the Milky Way Galaxy. This is the first time this term is ever used within the Dragon Ball franchise. If we use our own Milky Way galaxy as a standard, it comprises of approximately 250 billion stars or 2.5*10^11 stars (from NASA observation). Out of these stars, astronomers have found at least 2,500 stars that have orbiting planets so 2,500 solar systems. Again as we pointed out earlier, simply destroying a star would destroy neighboring planets and thus, the entire solar system as well. Hence, 2500 is really a lowball and 2.5*10^11 is the highball. Even taking the lowball, to destroy a galaxy, Kid Buu would have to be at least 2500 times more powerful than Super Perfect Cell. This does not seem logical. A Super Saiyan 3 Goku (Buu Saga) who is equal to Kid Goku is only 400x more powerful than a Base Goku that is WEAKER than Freeza’s fourth form. We know that Base Goku (BoG) < Freeza’s final form (120 million BP) because Beerus said so both in the Dragon Ball Super anime and manga. In the anime, it was directly stated when Goku powers up on King Kai's planet. In the DBS Manga, Beerus asks Whis why Goku has blonde hair and Whis responds that this is a Super Saiyan. Then, Beerus says he understands now how Goku defeated Freeza, which implies he did not understand before Goku turned into a Super Saiyan.
Source: Episode 5 of DBS Anime
Source: Chapter 2, DBS Manga
We know that SSJ2 Goku (Buu Saga) > SSJ2 Teen Gohan (Cell Saga). If SSJ2 Goku (Buu Saga) >= 1.25x SSJ2 Teen Gohan (Cell Saga), then SSJ3 Goku (Buu Saga) = 1.25x 4x SSJ2 Teen Gohan = 5x ~ 5x SSJ2 Teen Gohan. Then, if Kid Buu ~ SSJ3 Goku (Buu Saga) and SSJ2 Teen Gohan ~ Super Perfect Cell, Kid Buu >= 5x Super Perfect Cell. That would place Kid Buu only capable of destroying 5 solar systems (i.e. multi-solar system), not entire galaxies. One can even pick a higher multiplier than 1.25 between SSJ2 Goku > SSJ2 Teen Gohan, but it will get you nowhere near 2500 which is the needed number for Kid Buu to be a galaxy buster.
In DBS, the Elder Kaioshin states that the universe will be destroyed from the shockwaves generated from Beerus and SSG Goku's clashes is in danger from Beerus. This is the largest evidence in support that Beerus is a universe buster.
Source: Chapter 4, DBS Manga
In addition, it is stated in an interview of Toriyama that when Beerus sealed the Elder Kaioshin in the Z-sword, he could have destroyed the Kaioshin realm but it wasn’t appropriate. Since the Kaioshin realm is 1/10th of Universe 7, it means he is at minimum a 1/10th universe buster. The DB verse is different from the real world. The manga and guidebooks all imply that there are only four galaxies per universe - one for each Kaio to look after. There are about 100 billion galaxies in our own universe and 250 billion (2.5*10^11) stars approximately in each one. Since there are so few galaxies in the DB verse, I'll have to increase the number of stars in each one and assume that they are very large galaxies. So for this calculation, I will use 4 galaxies with 0.5 x 10^13 stars in each one which is about 100x bigger. This gives 2 x 10^13 stars in the DB universe approximately. If Super Perfect Cell is 1 solar system and Buu is 5 solar systems and Beerus in the log scale is a 10, which is 10^10, then Kid Buu is log (10^5 x 10^10) / (2 x 10^13) =~ 1.7 ~ SS3 Goku (Buu Saga). By the way, the entire log scale incorporates all this and is consistent with it. Note that this is only the destructive capacity to destroy 4 galaxies but not necessarily an infinitely expanding universe, but the shockwave from the resulting explosion of 4 galaxies will probably take out the rest just like SSG Goku vs Beerus's shockwaves almost did.
Oh, and I didn't use 1 Kili = 50,000 BP or Dabura's statement that 200-300 Kili is equivalent to a multi-planet buster because it is inconsistent. It puts Vegeta (18,000 BP) = 0.36 kili and Freeza (530,000 BP) = 10.6 kili.
Power level conversion
I have shown you guys how to derive the log scale and use it, but is there a way to convert this log scale to actual power levels? There have been no official power levels released since the end of the Freeza saga.
Use the excel spreadsheet here: https://www.dropbox.com/s/1wr2ll16h4rss ... .xlsx?dl=0
1. Akira Toriyama
2. Kanzenshuu & Herms98 (translations for interviews, Daizenshuu, and gathering of information): viewtopic.php?f=8&t=9633
3. Manga (DB, DBZ, DBS) - http://mangahasu.com/dragon-ball
4. GT Perfect Files
6. Anime (DB, DBZ, DBGT, DBS)
7. Hyourinjutsu and ygoofficial for sprites for transformation chart