The famous exponential equation 2^x=2x (ALL solutions)
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- čas přidán 13. 06. 2024
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0:00 Let's do some math for fun!
0:29 Review Lambert W function
2:28 Solve 2^x=2x
6:13 Why it looks like just one answer
9:27 Check out Brilliant
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We could actually get both answers from W(-ln(2)/2) by hand
See here: instagram.com/p/CSfIWPchpB1/?
I believe people don’t like it because it is clear that xe^x can not have well defined inverse because it is not biyective. Now sin(x) is not either, it’s just to properly define the argument and be sure one stays there.
Wow, that's an interesting way of going about it! Videos like yours inspire me to share my own maths tricks!
As I see you are a very good mathematician. I was working with a lot of equations back then! One of my favourite exercises like 10-15 years ago was the following. We have a and b where a,b e N! a^b+b^a = 423393 and a^a + b^b = 16780341. What is the value of a and b? Resolve it without just trying out numbers and hope we have luck!
Try solving this strange one(final version): lim d/dx -lg(2)/lg(1-1/x) as x is approaching infinity
No available 😢
I watched about 10 of your videos asking myself, "Why is this weirdo doing math while holding a PokeBall??" Then I finally saw one and realized it's your microphone with a cover on it.
Oh and i thought he has it just incase a random PI-kachu appears.
@@CrazyT2009😂😂😂
The pens are his wands and the pokeball is his pondering orb. This dude is an actual wizard of mathematics.
I don't understand much of any of this, but I really like your enthusiasm and way of teaching, the 10 minutes flew by before I even realised. Very entertaining channel
The Lambert W function is not an analytic function. Thus, one cannot present its formula using basic operations. (The sum and multiplication of analytic functions such as polynomials (The constant function is a special case of a polynomial.), exponential functions and trigonometric functions. (I may have omitted something.)
@@heinrich.hitzingermate why r u saying that here
@@egggames8059 Because he is secretly a genius. Real sigma males will understand.
Imagine bprp at the end of an epic video pulling out a green pen to finish it off!
Bprp: hmm new idea let's introduce rainbow pen too
@@gamin8ing Underrated comment😂
Noiceee
@@gamin8ingnoo, only straight education is needed...
I like how you fun about with math. it opens your mind to lots of possibilities.
Your grammar made my brain divide by 0.
@@shen144 Maybe because not everyone is a native english speaker ?
Steps:
6:58
You should also note that W0(- ln x/x) = -ln x for 0
X=1 and x=2 are easy solutions you can guess and then you can show that g(x)=2^x - 2x > 0 for x>2
So x=2 is the biggest solution
Then you can show that g(x)
Well, 9:27 is already in the video description so I have nothing to do this time 😂
I’m beginning to suspect the Lambert family is paying you every time you make a video mentioning the name.
😂
Here surveyors use the Lambert conformal conic projection. It's the same Lambert.
@@pierreabbat6157 man, I wish I could get in on some of that Lambert money.
For some reason no one wants to use the Winchell conformal tesseract mapping.
Recently there was a table "Derivatives For You" on the wall and now there is a painting "The Scream" by Edvard Munch. How are we to understand this?
"Maths for Fun" - "The Scream".
Pretty obvious, no?
he stuck the derivatives on his clothes so he no longer has that table on the wall
An claer and simple explanation of the two branches ! Thank you !
iˣ=2 then x=?
X = 2^i (i guess, i don't really know)
x=log(i)(2)
=ln(2)/ln(i)
=ln(2)/(πi/2 + 2πni), n is an integer
So the principle value is ln(2)/(πi/2), which is the same as ln(4)/πi
That's what I think.
@K.SRIKANTH REDDY MATHEMATICS yes, but that's exactly what I said, just slightly rearranged.
Wait, sorry. I am an idiot. My bad
log_i(2)
Because:
log_i(i^x)=log_i(2)
so log_i cancels out the i in i^x
Finally I was waiting for an explanation of the 2 branches!! Now I get it
Finally, the branches. The only thing we missed is how you don't need wolfram|alpha to figure out that -W_(0)(-ln2/2)/ln2 = 1. You coulda just gone with -ln2/2 = -ln2 · 2^(-1) = -ln2e^(-ln2)
Nice
Any ideas to find the -1 branch?
@@theuserings What do you mean?
I see, but what about -1 branch?
I am a class 11th students and I just got introduced to calculus few days ago and it's super interesting!!! Am more fascinated by the way this teacher switches to different marker in seconds!!!!!😳👍🏻
❤️ I love your videos! Thank You so much!
The fact that him making confused faces like he's geniuenly confused for teaching purposes is so hilarious 😂
I'm more confused at trying to decipher this in to English.. 😂
Thanks for the video
Okay okay I’ll subscribe already. Can’t believe you made math interesting
Where does the -1 in the "parameter" to the W function come from? What do the other values (not 0, 1) of that parameter represent when they are used? (Are they the complex roots of the original equation?)
Since the Lambert w function is an inverse function and it's not bijective you have to choose the branch. It happens to be that -1,1,0 are the easiest branches to work with. The intervals which the branches are are not consistent and the solutions are countably infinite. I suggest you look at the graph of the function and maybe that will clear up why.
Plz explain zeta function and riemann hypothesis 🙄
the last person who wanted to prove this in an open environment already died in January 2019 🙄
@@ymj5161 proving something and explaining what it is and what is states are two very different things...
@@anshumanagrawal346 lololol
@Castlier how are you calculating it I mean how did you know that it will converge at π²/6, is there any formula...
It's not the fact that this has two answers that surprises me. It's that the answer can produce integers, but have no analytic way to reduce it. Is there really no way to take your answer in the box and show those answers are 1 and 2 without approximating the W() function?
See the pinned comment
This is one beautiful problem that links the obvious 2 solutions of 2x=2^x and the 2 forms of the W function. I wonder if there is a possible generalization here beyond 2.
Your videos are very interesting
please kindly make videos on vector calculus.
5:44 "And that's a good place to stop."
Great video as always. What's Edvard Munch's The Scream doing in the background, tho?
😆
How to enter indices in W, in the Wolfiman calculator in the Lamberte formula?
I liked the graph of lambert W(x) function.
This question *_*exists*_*
Logarithm:- *Did anyone summon me?*
A different solution of this equation can be seen on my new channel called L+M=N at czcams.com/video/CC-L-OP71CM/video.html
Thanks sir 🙏
Hello blackpenredpen, how are you? Im sorry but I would like to program the lambert w function, can you help me? Is there a site to visit that could help me. Thanks so mucho for the content by the way, you are so smart! Salute you!
nice video, i liked it
Amazing!
We can generalize to a ** (x - y) = x ** z. , where y >= 1, a and z > 0.
The equation to study is f(x) = ln(x) / (x - y). If a > 1, there always two distinct solutions. If a 1, there is only one solution.
So ja great Video its so interesging. I'm finished my Abitur last Month but i Like to See thos Videos furthmore💅🤪🤖✨
This reminds me of how encapsulated funk takes place in real life and industries of skateboards.
Wow, that was an interesting way of going about it! Videos like yours inspire me to share my own maths tricks!
Bprp can u plz bring more content related to Recurance relations I'll appreciate it (at high school level) 😃😊
This was really fun!
Thanks bprp!
We can also log 2^x/2=x
X.lg2--lg2=lgx
Lg2(x--1)=lgx now remove log
2x--2=x
X=2
Thanks for your hard work 😸 i wish you good luck , greetings from Ukraine ))
very Nice.
You broke my mind when you multiplied both sides by -ln
hi bprp! is there a W-1 = f (W0)? in other words, is it possible to find W-1 having found Wo?
Do you mean:
Is there an f(x), such as f(Wₒ(t)) = W₋₁(t)?
In other words:
Is it possible to express W₋₁(t) using Wₒ(t)?
@@lukandrate9866 exactly
That I am not sure. Unless we have the vertical distance as what I pointed out in the video.
Fun fact tho, W1(-1/e)=W0(-1/e)=1
Is it possible to get an integral of 1-((x-1)/x)^x dx?
WolframAlpha just says it doesn't know.
I have a question about complex numbers :
If I have, m = a + bi & n = c + di , where a, b, c, d are real numbers and (i^2) = -1, then is,
n < m or, n > m?
The way to determine the `size` of complex numbers is to take their magnitude
M>N if |M|>|N|
|M| = sqrt(a^2+b^2)
|N| = sqrt(c^2+d^2)
@7:00
OK, so if W0 for the solution gives X = 1, that means that W0(-ln(2)/2) = W(ln(1/sqrt(2))) = ln 2
This is the first time I think I've ever seen you put the result of the W function into something that is not just a Wolfram numerical answer
Is there an analytical way to come up with that result?
You missed a minus sign, its actually -ln(2). Now the reason is, technically, you can rewrite -ln(2)/2 as -ln(2)*e^(-ln(2)), now see that this is in the form of xe^x, hence, W(-ln(2)/2)=-ln2. And also notice, if you multiply and divide by 2, we get -2ln(2)/4, which is -ln(4)e^(-ln(4)), hence W(-ln(2)/2)=-ln(4)=-2ln(2) if you restrict the range of W(x) to y
Sir I've been watching your videos and it really helped me develop interest in mathematics...earlier I scored 17/50 marks in previous maths test and now it's been 3 months the last test I got 48/50 and I'm the topper of my class.
Thank you Sir......
great!
Could u tell me, why we take n tends to infinity in limit where is infinity already undefined.
because we want to see what happens to the function as it gets closer to infinity
i just looked at it b4 he did the math and found 1 & 2 as solutions. After he did the math I had a mental breakdown
You make me love highschool maths, especially while I'm high
You are in love with lambert function🤩😍
Who isn’t? 😆
I want an approximation of Lambert W function with respect to other existing functions qwq
That lambert guy must have been a genius
When I do productlog equations I don't convert the number to base e first. I do it in the original base and convert to base e or whatever afterwards using this change of base formula.
W[base b](x)=W(x ln(b))/ln(b)
I think its much simpler
the equation 2ˣ= 2x can be solved in a simpler, graphical way: we plot y =2ˣ and y= 2x, after which we look at the intersections of the data with the graph and these points will be solutions to this equation.
therefore, x=1; x=2
I find it fascinating that such an innocent looking function as x(e^x) has a nose bleedingly crazy integral for its inverse.
I used to watch your videos in high school and couldnt understand a damn thing, now im in college studying cc and everything is clear now, mostly your calculus videos
Hey I have a pretty interesting question.can you solve this equation? "Logx(base a)=a^x”
Your awesome
An interesting generalization: a^x=a*x, 1=ax a^(-x)=ax e^(-x ln(a)),-ln(a)/a = -ln(a) x e^(-x ln(a)), so you have W(-ln(a)/a) in general. This means you have no real solution if -ln(a)/a0), one real solution if ln(a)/a=-1/e, two real solutions if -1/e
Gigachads: Graph the equations and find the common points
I wish I could double subscribe to you. You SUCH A GOOD TEACHER!!!
Thanks 😃
Assuming integers ... 1 & 2
Took about 2 seconds to work out in my head
Мне непонятно вот это уравнение:
W(x)*e^(W(x))=x
Откуда оно взялось?
~~~
I don't understand this equation:
W(x)*e^(W(x))=x
Where did it come from?
Math is the thing where when you’re learning something knew, if you look away for a second, you will be lost.
This is what I missed by not majoring in math in college? Chuck in a W. Chuck in a e. Chuck in a Log or a ln. 1 can be anything, 2 has no meaning. Then out of left field, tan, then sin of theta, the sec. Obvious.
The man on the painting shows his confusion 😂
it do has a simple way to solve it right.
1.20 why the second one is true?
What is ur hand sir
When I saw the title
My mind: x=2
Can someone tell me wether my approach also works:
2^x = 2x | :x
x^(-1)*2^x = 2
e^(-lnx)*e^xln2 = 2 | ln(…)
-lnx+xln2 = ln2 | :ln2
-lnx:ln2 + x = 1
-log_2(x) + x = 1 | +log_2(x), -1
x-1 = log_2(x) | (…)^2
x^2 -2x +1 = x | -x
x^2-3x+1 = 0,
and solving this is just a quadratic.
Would that be a valid solution?
Nvm, it isn‘t but where is the mistake?
Got exp(-W(-ln(2)/2) /2 when I did it. Results in the same results when evaluating in wolfram alpha so guess correct. But no clue how to reduce it to bprp solution without just doing his derivation 😞
How do you compute values of W_(-1)(x) by hand?
Use newtons method but pick x₁= some negative number
@@lukandrate9866 That's only an approximation, I wanna compute the actual precise values by hand
@@theimmux3034 It is like computing precise value of ln(5) by hand. It is impossible, the only thing you can do is to make a very accurate approximation
@@lukandrate9866 The precise value in the case of ln5 would be ln5 and it would be what I was looking for
@@theimmux3034 Ok so why you don't like the precise value of W₋₁(-0.23) as W₋₁(-0.23)?
You can tell if you wanna just express the lambert function without using the lambert function. Not just saying "I wanna a precise value". But I think W(x) is better than an infinite sum expansion or some other non-elementary functions
How to solve a slightly more difficult case 2^x=2x+5?
If reported that the original scream painting goes missing, we know who we'll be seeing😁
Question suggestion: x^2 - y^3 = 1, x and y are all integers, what are x and y? Note that there is only one answer for x and y, and you probably already found out x = 3 and y = 2
How does that work? (3)(2) - (2)(3) = 1???
@@weirdassbird I mean 3^2 - 2^3 = 1
Boom! I don't know Lambert W function has subscript, like wow that's how you define hidden number
We should have different bases for the W function like how we can have different bases for logs. The one issue I could think of is notation because W has multiple real branches (ln has multiple branches but it only has on real branch).
Maestro
I had a similar problem in school once
X^2 = 2^x
nice ;D
legends just do it by drawing graph
1 and 2 are the real answers that I got
Blue pen black pen red pen YAAAAAY!
Yes 😂👌👌
Please solve x²=2^x ❤
W LAMBERT function: the superoverpowered function
multiply both sides by x
x * 2^x = 2 * x^2
at this point.. idk lol
if the same input gives 2 outputs, it's no longer a function?
Well its exponential vs linear so u can just plug numbers till it stops working, 0 doesnt work, 1 works, 2 works, 3 doesnt and any number further wont either, hence answer is 1 and 2
The function f(x) = ln(x) / (x-1) , x > 0 , with f(1) = 1 is strictly decreasing and range all positive numbers.
Solutions for a**x = a*x , a > 0 . x = 1 is always a solution, if a > 1 , f(x) = ln(a) is the only one else.
I am feeling really stupid. How can x be 1 in that function?
@@stevendeans4211 Do you mean f(x)? If so, because the limit of f when x approaches 1 is also 1.
@@stevendeans4211 He specified f(1) = 1 at the discontinuity. He isn’t putting x = 1 in the function. For a > 1, f(a) = ln(a)/(a - 1) , f(2) = ln(2) but I don’t see the point. You can graph this function in Desmos in three parts: 0 < x < 1, x = 1, and x > 1. The discontinuity at x = 1 is removed by specifying f(1) = 1. (It is also true that f(a) = ln(a)/(a - 1) for 0 < a < 1.) At x = 1 the limit of the function from right and left has the form 0/0 so L’Hôpital’s rule applies. (The righthand limit is -1 as is the lefthand limit. If this is confusing, it is the fault of the terminology. A good reference is Olmsted’s Advanced Calculus)
@@honortruth5227 I get it. I misread the nomenclature. Thanks
Does this mean you can get infinitely many answers with any n?
Yes if you allow complex solutions.
what about n+n = n*n = n^n = n^^n = ... = n^...(infinite times)^n
I watched your video about Lambert W Function but I don't understand what the branch is. Now it's clear.
But if we input a number that is > 0 to W(x), then W-1(x) = W0(x) right ?
Không nha bạn ơi, cái nhánh -1 là một nhánh của hàm lambert w chỉ tồn tại với x trong khoảng [-1/e, 0). Nói cách khác, ye^y=x sẽ có hai nghiệm y nếu -1/e
@@waler1168 đúng rồi ngoài nhánh -1 và 0 thì toàn số phức thôi, cảm ơn bạn nha
You explained why there's two solutions, but can you explain why those particular n values are what we want, and even what the n values mean?
I can’t seem to find more info in that regard. So far I just know n=0 gives the principal branch (like the first answer) and n=-1 gives the other one (if any) on WolframAlpha.
Btw, any other n will give complex solutions which I have mentioned in my other videos like 2^x=x^2
@@blackpenredpen oh wow. I thought this was some sort of standard thing for branches