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
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
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 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
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
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 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! 🤣
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.
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
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.
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.
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.
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!!
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.
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.
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)
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.
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.
@@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.
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.
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
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 \( X \).
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.
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.
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!!
Never stop learning those who stop learning stop living, what great words with your ability to make math as simple as possible ♥
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
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 absolutely love how familiar yet abstract this problem feels. such a cool solution. thanks for sharing!
This is mind-blowing!!! 🤯
I am 72 and I´ve never heard something like this, tetration was unknown for me. always we can learn something new, Thank you.
the depth and breadth of your knowledge is amazing!
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.
Your videos always trigger the mathematician in me ❤❤
We need more teachers like him. And he should be idolized by many
Other than your amazing way of teaching and your enthusiasm, one more thig i like is the hats
You are a great teacher! Thanks for information.
You are the only one who think out of the box, Newtons. Superb Video, yet again!!😃
Чел лучший гетеросексуально логарифмирует! Пик и Эльмир гордятся тобой!
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
Craziest person on earth
Love you bro ❤
Solving maths to the next level
finally a teacher that doesn't unnecessarily overcomplicate things with the "a+b a/b (ab)" thing
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
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
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.
Man, your explanation are amazing ❤
After watching I am 40 years old again felling interst in mathematics. Thanks
The more I watch your videos the more inspired I get to learn more and more keep up the good work❤❤
i mean i knew how ²x had to be 2, but i got stuck after that so i continued the video
great explanation 👍
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.
These videos truly resparked my interest in mathematics, thank you
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 love explaining mathematics - thanks for your efforts
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
Thank you to teach us some technics with non trivial operations !
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
Wow ❤This is amazing. Take your flowers 💐 brother
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.
What an amazing guy!
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!
YOU*RE A CRAZY TEACHER! CONGRATULATIONS!
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.
Lovely, really nice to watch.
Very Nice class, Professor
Tks
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
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.
Make a video about every kind of exponential and tetration equations please
Nice example, Professor.
Tks
Thank you, from Iran 🤗
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!!
Just found this channel. Interesting
That was a great lesson!👍
Looks like Lambert even got to spend time with Euler. Pretty interesting! They knew about non-Euclidean geometry.
First Comment! You are a great teacher.
❤❤❤ I am a indian student verry nice
I didn’t know this channel!
This is amazing.
You... next level!
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.
haven't looked yet, from observation I think it's e^W(ln2)
Great explanation
Great stuff man!
Amazing...❤❤❤.
Great video
I did it with mental math, x^x = 2 , because 3^3 = 27 => x = e^(W(ln(2)))
You are already a pro
@@PrimeNewtons I'm an engineer so yeah in a way I'm pro lmao nice videos you're a great teacher!
Well explained sir!
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.
I wish if I had got one like you since I was in lower schooling system .. Ah !!!!!
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)
I i had a tutor like you in school then was to be excellent in mathy
Wow good Präsentation
Super..
This is heaven for me 👍, for the love of mathematics
Beautiful !
Superlog of both sides with base 3 could also eliminate the base 3 on both sides
Things I love!!!
We need more tetration problems
Great👍
It was super! 😮
Thank you! 👍
Super interesant! Excepțional! 🙏🎩✨🎗️💎 THANK YOU VERRY MUCH ! NEXT?! NEXT?!🌹🙏🎩
take superlog base 3 of both sides and you’re left with x^x=2
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.
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.
The derivative of the tetration function is as of 2023 not found, and an unsolved problem in mathematics!
@@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.
@@maxborn7400there's no elementary functions that express the integral of x^x. It simply does not exist.
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.
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
@@sportsloverbaseball I see my error. Thanks.
@@kimutaiboit8516 np :)
So interesting
nice video
incredible! you got yourself a new sub!
Welcome aboard!
Thanks, looking forward to what's ahead!@@PrimeNewtons
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 \( X \).
Take super square root by using Lambert w function
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
Solution = (sqrt(2))^(sqrt(2))
Thanks
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.
Never stop learning, because if you stop learning, you stop living
Thank God I'm in Med School 🙏🏻
a=b
Or, a/b=1
Or,(a/b)^1=(a/b)^0
So,1=0
Off course you can equate LH=RH at all times. In fact the question is 'solve for x in an equality.
√3
I swear the most two powerful tools in calculus are the natural log and Lambert W function🤣👍
Thank you
How parenthesis work in tetration, from up to down or vice versa