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.
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
@@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
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.
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. 🙂
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
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!
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.
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
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.
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
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.
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!
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.
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
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! 🤣
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
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.
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.
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?
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.
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.
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
@@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 👍
@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.
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
@@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
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
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.
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)
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.
You are not just a teacher, you are my inspiration to learn something new.
Thank for it❤
I have an A level mock tmr , this has genuinely helped me understand logs and exponentials better now thank you
Good luck
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
@@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
@@prateek1.9 some people enjoy math yk.
@@manyifung5411 I agree! This is fun but pretty useless especially in your age. Calculus it's a lot of better and extremely useful!!
I am 72 and I´ve never heard something like this, tetration was unknown for me. always we can learn something new, Thank you.
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.
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. 🙂
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
Never stop learning those who stop learning stop living, what great words with your ability to make math as simple as possible ♥
i absolutely love how familiar yet abstract this problem feels. such a cool solution. thanks for sharing!
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!
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
Man, your explanation are amazing ❤
the depth and breadth of your knowledge is amazing!
This is mind-blowing!!! 🤯
You are a great teacher! Thanks for information.
You are the only one who think out of the box, Newtons. Superb Video, yet again!!😃
Other than your amazing way of teaching and your enthusiasm, one more thig i like is the hats
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.
Craziest person on earth
Love you bro ❤
Solving maths to the next level
"Never stop learning." Always was my thing, + being curious of these stuff...
How am I, an average 16 year old suddenly finding maths so interesting and dwelling more on it??? Edit: Any INTPs?😀
Because math is great!
Here comes algebra...
Because most sciences are actually fun when you know what the hell is going on
Same but 15 year here
@@samthedjpro cooll! we can b friends!
The more I watch your videos the more inspired I get to learn more and more keep up the good work❤❤
Your videos always trigger the mathematician in me ❤❤
We need more teachers like him. And he should be idolized by many
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
but where does this W function even come from??
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.
finally a teacher that doesn't unnecessarily overcomplicate things with the "a+b a/b (ab)" thing
Чел лучший гетеросексуально логарифмирует! Пик и Эльмир гордятся тобой!
Wow ❤This is amazing. Take your flowers 💐 brother
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
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.
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!
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.
i mean i knew how ²x had to be 2, but i got stuck after that so i continued the video
great explanation 👍
These videos truly resparked my interest in mathematics, thank you
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
Very Nice class, Professor
Tks
After watching I am 40 years old again felling interst in mathematics. Thanks
Make a video about every kind of exponential and tetration equations please
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! 🤣
i solved it in my head too. took me like a minute since i am not really used to tetrations.
Lovely, really nice to watch.
I love explaining mathematics - thanks for your efforts
Thank you to teach us some technics with non trivial operations !
That was a great lesson!👍
What an amazing guy!
YOU*RE A CRAZY TEACHER! CONGRATULATIONS!
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
Nice example, Professor.
Tks
First Comment! You are a great teacher.
❤❤❤ I am a indian student verry nice
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
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.
How can we visualize 3 tetrated to the 1.56?
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
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.
Amazing...❤❤❤.
How can we solve if the tetration value is a fraction like ½2?
It must be a positive integer
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.
Working on it
Great explanation
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?
Great video
Just found this channel. Interesting
Well explained sir!
Thank you, from Iran 🤗
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.
Super..
Looks like Lambert even got to spend time with Euler. Pretty interesting! They knew about non-Euclidean geometry.
i figured it was the square root of 2 from the 'therefore' step, is this not correct? (1.41)
This is amazing.
How parenthesis work in tetration, from up to down or vice versa
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.
Great stuff man!
Where, in what science field, do we use tetration? What natural phenomena can be mathematically described by it?
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
Superlog of both sides with base 3 could also eliminate the base 3 on both sides
Is this comes in college level maths? Although I know how to solve it but I haven't learnt this in my school.
Thank you for this video.
We need more tetration problems
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 ?
i think that has yet to be defined
@@nzqarc I thought so, asked my professor he said you don't need to know this 😑
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.
@@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 👍
@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.
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?
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
okh bro still helpful tho :)@@freddier47
@@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
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
@@ryanman0083 actually, complex numbers have been defined too.
Great👍
I didn’t know this channel!
You... next level!
Pls explain about the pentation
hahahha ur like mee
What is 2 tetrated to 1/2?
Can we have a logical answer to this?
Wow good Präsentation
haven't looked yet, from observation I think it's e^W(ln2)
It was super! 😮
Thank you! 👍
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.
the answer is 2? (first question)
What is W
I have a doubt.. how would u solve this: ²2²? Which would u solve first the exponent or tetration?
That's an interesting question.
1.5?
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)
nice video
incredible! you got yourself a new sub!
Welcome aboard!
Thanks, looking forward to what's ahead!@@PrimeNewtons
Super interesant! Excepțional! 🙏🎩✨🎗️💎 THANK YOU VERRY MUCH ! NEXT?! NEXT?!🌹🙏🎩
I i had a tutor like you in school then was to be excellent in mathy
Hi how can I go about to solve
ˣ2=²ˣ7
J'ai "vaguement compris,", mais je ne comprends pas ce que représente W?
Et phi
Bro !!
I can't accept this solution until you define what is ˣa where x ∉ ℕ , please explain
x must be an integer
Beautiful !