Solving a tetration equation

[@The_Digital_Arts_World]

In all of the math courses I have taken through graduate school, I have never heard of tetration. This is absolutely amazing, the instructor is extremely powerful and exciting and yes, even at 71 years old … i learned something. I know I have used the Lambert W function before in math and engineering.

[@triggergaming7115]

You are not just a teacher, you are my inspiration to learn something new. Thank for it❤

[@aroushansenthilkumar]

I have an A level mock tmr , this has genuinely helped me understand logs and exponentials better now thank you

[@PrimeNewtons]

Good luck

[@Lordiustofu]

as a 13 year old, this video gave me a piece of mind of how math is really like, it isnt just numbers with the four operations nor sq roots, but it leads me to tetration, a whole new idea of how math works

[@prateek1.9]

@@manyifung5411 bruh why you 13 yr olds are getting into this complicated math thing ?you have a beautiful life to enjoy . also , you have to learn calculus anyways after 3 or 4 year later. why not enjoy now

[@Dojaesd]

@@prateek1.9 some people enjoy math yk.

[@ΔήμητραΜ-ψ1ρ]

@@manyifung5411 I agree! This is fun but pretty useless especially in your age. Calculus it's a lot of better and extremely useful!!

[@RootlessNZ]

Thank you for your instruction. I too have learned something new at 78 years of age. I have just come across tetration for the first time and I'm fascinated by how you manipulate the above equations so expertly. I'm hooked.

[@haroldlake1005]

I am 72 and I´ve never heard something like this, tetration was unknown for me. always we can learn something new, Thank you.

[@sobad2939]

Never stop learning those who stop learning stop living, what great words with your ability to make math as simple as possible ♥

[@LukieReal]

i absolutely love how familiar yet abstract this problem feels. such a cool solution. thanks for sharing!

[@dereklenzen2330]

I have a MS in mathematics, and I have actually never worked with the Lambert W function before. (Though I did use tetration once to rewrite a function raised to itself.) You have taught me something new! Thanks for a great video. Subscribing. 🙂

[@farmastam]

Can you teach some basics of mathematics . I am a high schooler and I need to get the unfair advantage before anyone else does . Or just you could recommend a bunch of maths videos that I should watch. Plzz

[@BetaLoversYT]

I am studying Economics at university, and, although this is the first time I see a video of yours, I feel I am going to use this eventually. Thanks for uploading it!

[@ZN9_xe]

Man, your explanation are amazing ❤

[@Xnes__1]

You are a great teacher! Thanks for information.

[@eq.kiraaa]

Ok the fact that this guy is so talented he's making me feel like I can understand whatever the hell he's writing blows my mind

[@christopherrascon6386]

This is mind-blowing!!! 🤯

[@JaseewaJasee]

the depth and breadth of your knowledge is amazing!

[@rajibchaudhury7974]

You are the only one who think out of the box, Newtons. Superb Video, yet again!!😃

[@psevdonim753]

Чел лучший гетеросексуально логарифмирует! Пик и Эльмир гордятся тобой!

[@sonicwaveinfinitymiddwelle8555]

For moment I thought the question was wrong due to 2 superexponents in row but luckily this video explains how to really solve tetrational problem. Something new learned today.

[@sourishnanda1809]

Craziest person on earth Love you bro ❤ Solving maths to the next level

[@enzoys]

but where does this W function even come from??

[@budderman3rd]

It's the inverse function of xe^x. Mathematician found there is no way to get x from xe^x, so they figured out what the inverse function would be and how it worked and called it lambert w function or product log. It became a useful function for solving certain equations.

[@parigupta86011]

Your videos always trigger the mathematician in me ❤❤

[@williamkeitaro8910]

finally a teacher that doesn't unnecessarily overcomplicate things with the "a+b a/b (ab)" thing

[@Kobe24SherlockHolmesChaitanya]

Other than your amazing way of teaching and your enthusiasm, one more thig i like is the hats

[@ibrahimyazici5651]

We need more teachers like him. And he should be idolized by many

[@devydanger]

"Never stop learning." Always was my thing, + being curious of these stuff...

[@yasinforughi-b1z]

The more I watch your videos the more inspired I get to learn more and more keep up the good work❤❤

[@LucSzI]

These videos truly resparked my interest in mathematics, thank you

[@davestillhere4169]

Lovely, really nice to watch.

[@gilmartrevisan]

Very Nice class, Professor Tks

[@onyx_enigma]

i mean i knew how ²x had to be 2, but i got stuck after that so i continued the video great explanation 👍

[@gunssom0078]

I got to ²x = 2, but didn't know Lambert W Function and couldn't solve it. Will watch that Video. Thanks for teaching this in a Clear and Straightforward way

[@markfekete5078]

You have all the passion of the world and I really respect that. thx for this equation solving. I'm not a big fan of math but your presentationvwas really great.

[@Khaled_HR13_kardashev]

What an amazing guy!

[@redroach401]

Omg i just came from your other video about x to the super power 2 equals 16 and used the same method to solve this and it took less than a minute. Thank you soooo much for yraching me this cool trick

[@scarletevans4474]

I tried to solve it in my head and ended up equating tetration with exponentiation, getting log(3) instead of log(2)... then, I immediately remembered what you said about our brains, at the beginning of the video! 🤣

[@heco.]

i solved it in my head too. took me like a minute since i am not really used to tetrations.

[@VanshikaRathod-v5b]

You sir is great 🎉🎉 best thanks to your teaching skills, whenever i saw your video my reaction like 😮, thanks sir keep it up if you not make video my reaction like 😢, sorrry ok bye😅, thansk for 10 million likes to this comment

[@arminklar3613]

Wow good Präsentation

[@dannygolds]

I'm studying in Russia and I think we have so little knowledges for W-function, or our lecturers just dont' wanna to learn to us with it, jush perfect solution we suggested, master!

[@heco.]

You're absolutely right! We are learning so many complex algebra equations but not the LambertW function. And because of that we can't solve stuff like x^x = 2. I am in my last year of high school but we don't have LambertW function in college. So, I learned it entirely from internet.

[@goabaonesekalaba2830]

Wow ❤This is amazing. Take your flowers 💐 brother

[@gilmartrevisan]

Nice example, Professor. Tks

[@bigobibishop]

Great video

[@kumar1036]

Great explanation

[@ExpungedFunkyFriday]

Make a video about every kind of exponential and tetration equations please

[@Mehrdad_Basiry-fj4rl]

Amazing...❤❤❤.

[@carlossantosalmeida6396]

YOU*RE A CRAZY TEACHER! CONGRATULATIONS!

[@elishagunasekara4038]

That was a great lesson!👍

[@abumarwan6]

I love explaining mathematics - thanks for your efforts

[@vorpal120]

This was great! Thanks for sharing some under taught maths. No one ever showed me this stuff. I'm just playing here but, now for the sarcasm: Never stop learning? Those who never stop learning, forget. Those who stop learning, remember. Meaning I got a finite amount of memory and the more I cram in my head nowadays I tend to lose something else. But if I hold on long enough to what I know. I will remember those memories, longer. I just forgot where I put my keys.... bummer.

[@michaelbaum6796]

Great👍

[@corbonmaths9597]

Thank you to teach us some technics with non trivial operations !

[@kripaasp]

After watching I am 40 years old again felling interst in mathematics. Thanks

[@stevenHUH]

Well explained sir!

[@urs2357]

Super..

[@aminniku6898]

Thank you, from Iran 🤗

[@sudiptoatutube]

First Comment! You are a great teacher.

[@aryatripathi06]

How am I, an average 16 year old suddenly finding maths so interesting and dwelling more on it??? Edit: Any INTPs?😀

[@HomieSeal]

Because math is great!

[@Bruh-bk6yo]

Here comes algebra...

[@mrchewy679]

Because most sciences are actually fun when you know what the hell is going on

[@samthedjpro]

Same but 15 year here

[@aryatripathi06]

@@samthedjpro cooll! we can b friends!

[@jbot8108]

nice video

[@infraredaustrian6714]

You... next level!

[@ryanchicago6028]

Looks like Lambert even got to spend time with Euler. Pretty interesting! They knew about non-Euclidean geometry.

[@JCSol]

This is amazing.

[@RajeshSharma-zm8et]

❤❤❤ I am a indian student verry nice

[@ГригорийФилипп-п9у]

Since 3^3=27, 3^^2 = 27 (^^ equals tetration), and we should find x that, been tetrated to 2, gives us a 2, so it is just x^x=2

[@Necrozene]

Great stuff man!

[@HeidiRobinson-q6d]

Thank you for this video.

[@tyal-q9x]

Just found this channel. Interesting

[@goabaonesekalaba2830]

I didn’t know this channel!

[@dilshad003]

Beautiful !

[@PlatonAltei]

So interesting

[@2106522]

It was super! 😮 Thank you! 👍

[@AbouTaim-Lille]

Can you give more detalis on the Lambert W function. If it is just another special function it is ok but we need Its , properties, its division and integral. And Its taylor series.

[@PrimeNewtons]

Working on it

[@nikkirolle7032]

Things I love!!!

[@JOEAMV1]

Pls explain about the pentation

[@aryatripathi06]

hahahha ur like mee

[@marklyles3335]

Can 3/2 also be correct? I don't know Lambert's Function. I really need to brush up on logs and learn about e.

[@フミズ]

How can we solve if the tetration value is a fraction like ½2?

[@PrimeNewtons]

It must be a positive integer

[@Mech141]

x is 1 because 3 raised tetrated by 2 is equal to 3 to the power of 3 which is 27 and 3 tetrated by 1 is 3 so x=1

[@venkybabu8140]

Super power is somewhat like double integrals and triple etc. because a super power series can always be replaced by integrals. Base depends on grains.

[@igravity2965]

How can we visualize 3 tetrated to the 1.56?

[@tjahyotamtomo7189]

The difference between the solutions you mentioned-\( X = \sqrt{3} \) (approximately 1.732) and \( X = 1.56 \)-suggests that something might have gone wrong in the logarithmic approach that resulted in the value \( X = 1.56 \). Let’s carefully revisit the logarithmic approach to understand where any discrepancy might have arisen. ### Re-examining the Logarithmic Approach The equation we are solving is: \[ 3^{X^2} = 27 \] 1. **Step 1: Take the natural logarithm of both sides:** \[ \ln(3^{X^2}) = \ln(27) \] Applying the logarithm power rule, \( \ln(a^b) = b \cdot \ln(a) \): \[ X^2 \cdot \ln(3) = \ln(27) \] 2. **Step 2: Express \( \ln(27) \) in terms of \( \ln(3) \):** Since \( 27 = 3^3 \), we can write: \[ \ln(27) = \ln(3^3) = 3 \cdot \ln(3) \] 3. **Step 3: Solve for \( X^2 \):** Substituting \( \ln(27) = 3 \cdot \ln(3) \) into the equation: \[ X^2 \cdot \ln(3) = 3 \cdot \ln(3) \] Dividing both sides by \( \ln(3) \) (which is a positive number): \[ X^2 = 3 \] 4. **Step 4: Solve for \( X \):** Taking the square root of both sides: \[ X = \sqrt{3} \approx 1.732 \] ### Addressing the \( X = 1.56 \) Result If you obtained \( X = 1.56 \), this might indicate: - **An approximation issue** during intermediate steps. - **A calculation mistake** in how the logarithms were applied. - **A different interpretation** of the equation, which led to a slightly altered result. ### Conclusion The correct solution using either the pattern recognition approach or the correct logarithmic method should yield \( X = \sqrt{3} \approx 1.732 \). This matches the expected result when solving \( 3^{X^2} = 27 \). If you consistently get \( X = 1.56 \) using the logarithmic approach, there may have been a mistake in one of the logarithmic steps. Double-check the logarithmic steps to ensure they correctly follow from the given equation.

[@NameUV3]

1.5?

[@mrsadmangaming6350]

um bro what if the tetration is in the negetive so i mean that what if it was said that 3 raised to the super power -2 . how would u define it?

[@freddier47]

I remember reading somewhere that we haven’t defined tetration for non-natural numbers yet but I can’t remember where I read that so that could be a complete lie

[@mrsadmangaming6350]

okh bro still helpful tho :)@@freddier47

[@ryanman0083]

@@freddier47Real numbers are defined We can understand better with super Logarithm (inverse of Tetration) By definition sLog2 (2^^3) = 3 NOTE: "sLog" is a notation for super Logarithm. Like how Logarithm cancels the base leaving the exponent ex. Log2 (2^3) = 3 super Logarithm does the same with Tetration leaving the super power. We can use super Logarithm to solve non integer super powers since super Logarithm is repeated Logarithm by definition. Let's let sLog2 (16) = 3+x Where 0 ≤ x < 1 (represents a 0 or decimal) sLog2 (2^^3) = sLog2 (2^2^2) => Log2(2^2^2) = 2^2 => Log2(2^2) = 2 =>Log2(2) = 1 At this point we've taken three logs representing our integer part of the solution (given by the fact that the answer is equal to 1). We just take log again for the decimal x (the remainder of 2's that we need.) Log2 (1) = 0 Thus sLog2 (16) = 3+0 = 3 Well let's look at what happens when we go backwards through the same process to see what happens to the remainder. Log2 (Log2 (Log2 (Log2 (16)))) = 0 Log2 (Log2 (Log2 (16))) = 2^0 Log2 (Log2 (16)) = 2^2^0 Log2 (16) = 2^2^2^0 16 = 2^2^2^2^0 = 2^2^2 = 2^^(3+0) The remainder adds an extra '2' to the top of the power tower and the additional 2 is raised to the power of the remainder For 0 ≤ x ≤ 1 By definition sLog a(a^^3+x) => a^a^a^a^x By definition of Tetration a^^3+x = a^a^^2+x = a^a^a^^1+x = a^a^a^a^^x a^a^a^a^^x = a^a^a^a^x a^a^a^^x = a^a^a^x a^a^^x = a^a^x a^^x = a^x by definition for 0 ≤ x ≤ 1 We can take for example 3^^e ≈ 3^3^3^(e-2) ≈ 227,914.4 It's complex numbers (i.e. 2^^i) that it's not defined

[@ryanman0083]

For real numbers a^^x = a^x for 0 ≤ x ≤ 1 for negative super powers we use the above definition to extend to negatives -1 ≤ x ≤ 0 a^^x = Log a (a^^x+1) NOTE: because -1 ≤ x ≤ 0 then 0 ≤ (x+1) ≤ 1 thus the above rule applies a^^x = Log a (a^^x+1) = Log a (a^x+1) = (x+1)*Log a(a) = x+1: For -1 ≤ x ≤ 0 a^^-2 is undefined because a^^-2 = Log a(a^^-1) and a^^-1 = 0 by definition so a^^-2 = Log a(0) and you can't have 0 in the Log

[@nzqarc]

@@ryanman0083 actually, complex numbers have been defined too.

[@Thebeastv01]

hello sir greetings can't we do like this from this step i.e. (2^)X = 2 mentioned in the video (2^)X = (root 2)^2 x = (root 2) x=1.414 is this correct, if not can you provide the reason why it is wrong

[@ericcarterofthehillpeople]

So 3 to the superpower of 1.56 is 3 raised to the third power ... 1.56 times? How does that calculation work? If I wanted to do it by hand, how would I do that?

[@islammuradov1]

❤❤❤❤❤

[@cosmicgarou99]

This is heaven for me 👍, for the love of mathematics

[@CC--qn4gf]

We need more tetration problems

[@samthedjpro]

How we can build exponents with decimal numbers , like x superpower 3 is ((x power x) power x) but what would be x superpower 1.56 ?

[@nzqarc]

i think that has yet to be defined

[@samthedjpro]

@@nzqarc I thought so, asked my professor he said you don't need to know this 😑

[@kimutaiboit8516]

I imagined such a problem. To keep it simple let's start with ½^X That will be 1/(²X) Now for 1.5^X it will be ³X/²X. Just my thoughts.

[@samthedjpro]

@@kimutaiboit8516 I also thought that but I was wrong as x time x is x² or x^2/1 and x^1/2 will be opposite which is how many times it should be multiple so it makes x which is √x but when we talk superpowers (supeman 😂) ²x will means x power x but x superpower 0.5 must means how many times x should be exponentiate to make x which neither logs or any function can define, if you have any answer please comment 👍

[@kimutaiboit8516]

@samthedjpro let's say x=3 ²3 is 3³ ¹3 is 3 My claim was ½3 is 1/3³ But now that you mention it, I think I have to relook it. The expression I made is for -²3 I think ½3 is undefined as you said. 🤔 PS: What could ⁰3 be? I guess undefined too.

[@_Diana_S]

Where, in what science field, do we use tetration? What natural phenomena can be mathematically described by it?

[@Wolfy-pw2py]

Mathematician obviously since they deal with all maths related and probably math that involves studying space since the universe is infinite and involve big numbers like from billion up to sextillion and more light years

[@Aerobrake]

incredible! you got yourself a new sub!

[@PrimeNewtons]

Welcome aboard!

[@Aerobrake]

Thanks, looking forward to what's ahead!@@PrimeNewtons

[@DSN.001]

How parenthesis work in tetration, from up to down or vice versa

[@maxborn7400]

finally getting crazy here lol. Differentiation/Integration of a variable tetration would be interesting. I think it's more complicated than the x^x^x^x you did some time ago.

[@aguyontheinternet8436]

Well since the function isn't continuous, only defined for positive integers, you'd have to do something _real_ creative to differentiate/Integrate something like 2↑↑x, but it would probably be easier to find a way to extend the function to the reals.

[@francaisdeuxbaguetteiii7316]

The derivative of the tetration function is as of 2023 not found, and an unsolved problem in mathematics!

[@maxborn7400]

@@aguyontheinternet8436 I would imagine continuation to at least real number line is not an issue: logarithms are known, no matter what "degree" of exponentiation, so something like 1.234^3.678^6.845 is technically defined. It's a mess to write it in tetration form, but something like x↑↑ x etc is defined (Idk how to write tetration on keyboard, but I am abusing notation from Graham's number to get the point across). So technically, a tetration of variable real numbers is defined and continuous.

[@flightyavian]

​@@maxborn7400there's no elementary functions that express the integral of x^x. It simply does not exist.

[@אבאאבא-ע1מ]

Superlog of both sides with base 3 could also eliminate the base 3 on both sides

[@LaurentiuSbera-nf3wr]

Super interesant! Excepțional! 🙏🎩✨🎗️💎 THANK YOU VERRY MUCH ! NEXT?! NEXT?!🌹🙏🎩

[@A.A.A543]

Is this comes in college level maths? Although I know how to solve it but I haven't learnt this in my school.

[@neevee_gd]

the answer is 2? (first question)

[@ModelTypeZero]

i figured it was the square root of 2 from the 'therefore' step, is this not correct? (1.41)

[@crowreligion]

What is 2 tetrated to 1/2? Can we have a logical answer to this?

[@mzzsrifkfk1271]

What is W

[@helixfire1198]

I never learned lambert functions. is that calculus ? Also i barely remember log functions on my texas instrumemts calculator. Im going to watch videos on this. I wish this youtube channel was around years ago when i was in school.

[@FELISCATY]

I i had a tutor like you in school then was to be excellent in mathy

[@ArokoyoEsther-zo4nd]

Thanks