If you assume your audience knows what Lamberts W function is without any further explanation, I would think you can safely assume you don't have to spell out the initial divisions by x and 2 in painstaking detail.

Why is W more valid than J? J is a function I just made up which is the inverse of the function x^(-2)*8^x. So the answer is J(1). Easy

WRONG! You have to do 3 cases : 1) If x>0 : ln(x²) = 2*ln(x) and x = exp(ln(x)). Then, you have an equation without any solution since -ln(8)/2 < -1/e, so impossible to apply W on both sides. 2) If x

At the beginning is the equality ln(x^2)=2ln(x) which is correct only when x>0 and at the end this calculation gives x=--0,559..That does’nt seem rigorous, even if the solution is right.

Lambert W function: "Then a miracle occurs."

Good point!

W is an approximation so you could might as well just iterate the solution. it like solving x^x = 12

Thanks sir

or you could use the constant of 1²= 9.87. so x = .5559. half of 1 if you include time

The question is awesome.

I wonder if Lambert knew he'd become the most trendy math trick on KZfaq.

There is a big mistake in very first step. it should 2ln(|x|) not 2ln(x). your negative fraction result won't hold otherwise.

Honestly I would have just solve it graphically from the beginning. Then I would have tried to insert some values in the beginning equation to see what’s the interval in which X lays. For example I would have tried values like 0, 1, -1, -2 and so on. I would have gotten a interval and not an exact value, of course. But at least it would have been much easier and faster

The lambert W function is so disappointing :/ I can't explain it but I always feel let down when it's that function

this is like a textbook trivial application of lambert-w. Why did you even make this video if THAT was the thing you wanted to show? Was the purpose of this to just raise awareness of the lambert-W function? What on earth does this have to do with the IMO?

I don't think you've answered the question. If you can just denote any zero of the function ln(x)/x - a by Xa and then tell the solution as a function of Xa, then you have not really answered the problem !

Using Numerical Analysis

From graph plot x should be around -0.6. I have PhD in physics but never heard about Lambert function.

It’s higher maths in university. Not for year 12.

I think the creator of this video missed the point of this problem: you have to show that there is no closed-form solution for problems like this. Actually, you can only write the solution with Lambert's W function or in another way using functions that can't be expressed with a closed-form expression. This is the important thing you have to learn about problems, where a variable is in the basis and exponent at the same time.

🙏🌹😍🙃😃 NEXT ?!

You can't use the rule ln(x^2)=2 ln(x) for x < 0.

Love the explanations thanks for your video ❤ just one feedback it’s not said “NAYtural” - Although the word Nature is distinctly so, the Na in ‘NAtural’ is pronounced same way as NASA, or Na X.. in russian 😇

Lambda w function? Never heard of this

Who is Double You?

I thought it would be over at 2:11; lnx=1/x and over x = ln8/2. But the x would disappear and there would be no solution to it. I think this eqaution is not solvable.

My biological science ass would just graph and find where they intersected. I like math and I took all the way to calculus 3 at the college level but I honestly never have to use it anything except algebra in my job.

I have read that the Lambert function can be solved only if x is greater than -1/e, which is ≈ -0,367879. But your solution x ≈ -0.559 is lower.

You didn't discuss the conditions where x could be equal to or greater or lesser than zero.Sad.

Bro’s l’s and e’s look exactly same lol (but fr it made it more confusing)

lnx on a negative x?

Because (-ln8/2) is less than (-1/e), W(-ln8/2) is a complex number = -0.291 + 1.36 i. Can you show how you solved for x? e^(real number) cannot be negative.

Is W(z) transcendental?

على ماذا تدل W وكيف علمت قيمتها

Ln = loge X , sendo Ln X = a ; E e^a= x

Nice, clever and all that... but how can anyone write x like that 😭

The universe opened up when this guy took out his red pen

Why do you say this is olympiad question??

To use the W function is not solving

Ne comprenant pas l'anglais, je vais devoir repasser cette vidéo en plusieurs fois... ça va être comme à la télé : des rediffusions.

Why in the hell would anyone write an X like that? That is the most annoying thing I've ever seen.

One has infintely many complex solutions , using the W-function and its complex extensions .

That 'ln' kinda looks like 'e sub n' just saying. At a glance it would appear confusing to the uninitiated Math student....but nice methodology all the same!

Это не красивое уравнение. И не красивый ответ Просто посмотрев на него становится понятно, что х отрицательная величина. И она приблизительно -1/2 . Решил?

x = 8 and 2 = x. ez

I'm not a fan of this method. Seems complicated/busy. But thank you for the lesson anyway.

This is the pinnacle of a smart person thinking they are explaining something simply. Try again please, a little bit slower.

Uhh I solved it in 1 min tho? Am i wrong? 8×8×8×8=64×64 Simple's the best- by a 11yo

x=0

0

I understood nothing.

meaningless...

Cosa sarebbe w ()???