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.

[@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!

[@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.

[@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 ❤

[@JaseewaJasee]

the depth and breadth of your knowledge is amazing!

[@Xnes__1]

You are a great teacher! Thanks for information.

[@psevdonim753]

Чел лучший гетеросексуально логарифмирует! Пик и Эльмир гордятся тобой!

[@devydanger]

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

[@christopherrascon6386]

This is mind-blowing!!! 🤯

[@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.

[@ms_ks4]

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

[@ibrahimyazici5651]

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

[@rajibchaudhury7974]

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

[@sourishnanda1809]

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

[@Kobe24SherlockHolmesChaitanya]

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

[@williamkeitaro8910]

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

[@yasinforughi-b1z]

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

[@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

[@LucSzI]

These videos truly resparked my interest in mathematics, thank you

[@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.

[@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.

[@davestillhere4169]

Lovely, really nice to watch.

[@gilmartrevisan]

Very Nice class, Professor Tks

[@parigupta86011]

Your videos always trigger the mathematician in me ❤❤

[@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!

[@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.

[@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

[@abumarwan6]

I love explaining mathematics - thanks for your efforts

[@kripaasp]

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

[@gilmartrevisan]

Nice example, Professor. Tks

[@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.

[@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.

[@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 👍

[@corbonmaths9597]

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

[@carlossantosalmeida6396]

YOU*RE A CRAZY TEACHER! CONGRATULATIONS!

[@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

[@goabaonesekalaba2830]

Wow ❤This is amazing. Take your flowers 💐 brother

[@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.

[@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

[@ExpungedFunkyFriday]

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

[@kumar1036]

Great explanation

[@Khaled_HR13_kardashev]

What an amazing guy!

[@bigobibishop]

Great video

[@elishagunasekara4038]

That was a great lesson!👍

[@Mehrdad_Basiry-fj4rl]

Amazing...❤❤❤.

[@ryanchicago6028]

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

[@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

[@stevenHUH]

Well explained sir!

[@urs2357]

Super..

[@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.

[@michaelbaum6796]

Great👍

[@arminklar3613]

Wow good Präsentation

[@RajeshSharma-zm8et]

❤❤❤ I am a indian student verry nice

[@aminniku6898]

Thank you, from Iran 🤗

[@sudiptoatutube]

First Comment! You are a great teacher.

[@Necrozene]

Great stuff man!

[@tyal-q9x]

Just found this channel. Interesting

[@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

[@infraredaustrian6714]

You... next level!

[@JCSol]

This is amazing.

[@jbot8108]

nice video

[@goabaonesekalaba2830]

I didn’t know this channel!

[@mrblakeboy1420]

you can easily see that you have to solve it as 3^^(x^x)=27 , which means you have to solve it as x^x=2. i don’t have enough math experience to solve it from there, probably something with e^ln(x^x)

[@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.

[@dilshad003]

Beautiful !

[@2106522]

It was super! 😮 Thank you! 👍

[@PlatonAltei]

So interesting

[@jonathanmulenda8668]

I did it mental math without the Lambert fnctn: sqroot of 27=5.1962 , because 27 is what we get after the last super power then remained with 3^x=5.1962...x=logbase3(5.1962) and x=1.5 sorry I used a lot of space..... You're doing great cudos!!

[@igravity2965]

How can we visualize 3 tetrated to the 1.56?

[@フミズ]

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

[@PrimeNewtons]

It must be a positive integer

[@Aerobrake]

incredible! you got yourself a new sub!

[@PrimeNewtons]

Welcome aboard!

[@Aerobrake]

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

[@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.

[@_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

[@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.

[@tjahyotamtomo7189]

Let's break this down into simpler terms to make it clearer for a layperson: ### The Problem: You have an equation that looks like this: \[ 3^{(X^2)} = 27 \] This means: "Three raised to the power of \( X^2 \) equals 27." ### Goal: We need to figure out what \( X \) is. ### Two Approaches to Solve the Problem: #### 1. **Pattern Recognition (Quick Way):** - **Step 1:** Recognize that 27 can be written as a power of 3. Specifically: \[ 27 = 3^3 \] - **Step 2:** This lets us rewrite the equation as: \[ 3^{(X^2)} = 3^3 \] - **Step 3:** Since the bases (the number 3) are the same on both sides, we can focus on the exponents. So: \[ X^2 = 3 \] - **Step 4:** To find \( X \), take the square root of 3: \[ X = \sqrt{3} \approx 1.732 \] #### 2. **Logarithm Method (Longer Way):** - **Step 1:** Start with the same equation: \[ 3^{(X^2)} = 27 \] - **Step 2:** Use a technique called logarithms to make the equation easier to handle. Taking the logarithm of both sides helps to bring down the exponent \( X^2 \): \[ \ln(3^{(X^2)}) = \ln(27) \] - **Step 3:** This allows you to pull the \( X^2 \) down in front of the logarithm: \[ X^2 \times \ln(3) = \ln(27) \] - **Step 4:** Recognize that \( \ln(27) \) is the same as \( 3 \times \ln(3) \). So the equation becomes: \[ X^2 \times \ln(3) = 3 \times \ln(3) \] - **Step 5:** Divide both sides by \( \ln(3) \) to simplify: \[ X^2 = 3 \] - **Step 6:** Take the square root of 3 to find \( X \): \[ X = \sqrt{3} \approx 1.732 \] ### Why the Difference in Results? If someone got a different answer like \( X = 1.56 \), it might be because of a mistake in the logarithm steps or an incorrect approximation. The correct answer, whether using pattern recognition or logarithms, should give you: \[ X = \sqrt{3} \approx 1.732 \] This is the correct value for \( X \).

[@cosmicgarou99]

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

[@anthonypierzchala-vi8en]

haven't looked yet, from observation I think it's e^W(ln2)

[@Hanible]

I did it with mental math, x^x = 2 , because 3^3 = 27 => x = e^(W(ln(2)))

[@PrimeNewtons]

You are already a pro

[@Hanible]

@@PrimeNewtons I'm an engineer so yeah in a way I'm pro lmao nice videos you're a great teacher!

[@CC--qn4gf]

We need more tetration problems

[@MOVIEPARADISE12]

Pls explain about the pentation

[@aryatripathi06]

hahahha ur like mee

[@nikkirolle7032]

Things I love!!!

[@aryaprasad9974]

I have a doubt.. how would u solve this: ²2²? Which would u solve first the exponent or tetration?

[@PrimeNewtons]

That's an interesting question.

[@אבאאבא-ע1מ]

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

[@nancymatro8029]

Please explain in detail what you mean by superpower. Can't find anything about your notation using superscripts on the left, nothing even in the lengthy Wiki article on tetration in which they have tables of different notational styles, but none like yours.

[@GodGapple]

Never stop learning, because if you stop learning, you stop living

[@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.

[@ModelTypeZero]

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

[@CatInABaseballCap]

take superlog base 3 of both sides and you’re left with x^x=2

[@LaurentiuSbera-nf3wr]

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

[@FELISCATY]

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

[@Stn_4]

I wish if I had got one like you since I was in lower schooling system .. Ah !!!!!

[@kimutaiboit8516]

Someome asked about 1.5^X 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.

[@sportsloverbaseball]

I might be misunderstanding, but 1.5^x≠x tetrated to 1.5. It would be 1.5 to the power of x. For tetration, use ^^. For example, the equation is 3^^x^^2=27

[@kimutaiboit8516]

@@sportsloverbaseball I see my error. Thanks.

[@sportsloverbaseball]

@@kimutaiboit8516 np :)

[@TheMathManProfundities]

A word of caution here, you explained the original left hand side as being ²(3↑↑x) when in fact it is 3↑↑(²x). You demonstrated this correctly when you equated 3↑↑(²x) to 3↑↑(2) but your expansion was wrong.

[@TheMathManProfundities]

Typo: meant to say explanation not expansion