Germany - Math Olympiad Exponential Problem.

Germany - Math Olympiad Exponential Problem.

[@maciejterakowski9062]

It is rather "to guess", not "to solve"

[@K.Klogic]

Thank you for your comment ❤️💖

[@jb.drills]

This why enterpreneurs are good on arithmetics but not good at algebra

[@mathophilechirag]

kzbin.info/www/bejne/m4G5e3-vfbWhoLMsi=nIJMdBlYbU0npQMu

[@urwahhassoon6886]

there is something called “mathematical induction” it is legit

[@張三豐-l4v]

這確實不是好的解答

[@vvv1041]

This method isn’t analytical. For instance, it doesn’t answer if this equation has only one solution. For the similar equations like that but with different coefficients it will not work.

[@K.Klogic]

Thank you for your comment ❤️💖

[@jursamaj]

Indeed, there are 2 other real solutions, plus multiple families of complex solutions.

[@sergeykardash4482]

​​@@jursamaj1 more real...

[@zorg6020]

Есть и проще решение чем то что в видео. Найти решение на пересечении графиков функций. Вот и все.

[@cristinap.7498]

A me basterebbe trovarne una

[@timmy18135]

Just use logarithm

[@progr6171]

Geterosexual logarithm*

[@saturninufolud6767]

You would receive something with the Lambert W function (irrational in general)

[@progr6171]

mу соmmеnt wаs dеlеtеd.... It sаid: hеtеrоsехuаl lоgаrithm

[@gmddlex5873]

​@@progr6171спалился

[@AW-dt8ct]

​​@@progr6171 And is there the homosexual one?

[@howardhavlicsek5031]

Doesn't look like anyone is willing to solve the problem, just guess at a solution! What if the problem was 2^x = x^31?

[@dhy5342]

Then X would equal 1.0231407...

[@K.Klogic]

Thank you for your comment ❤️

[@K.Klogic]

Thank you for your comment ❤️💖

[@MKPoqdPDeamimzpygxDjg7sJY59]

@@dhy5342 or about 246.266

[@jursamaj]

@@dhy5342 Also 246.266…, plus 5 families of complex answers.

[@oguzkk]

Solution set of This equation has three real roots. One of them is 256. in the question, it has to be stated that x was an integer.

[@manassarma735]

I must be super drunk off my ass to have watched the whole thing

[@markharder3676]

Take a guess and assume that like 32, the answer is a power of two. Use logarithms in base 2 because log(2, 2^x) = x. and log(2, x^32) = 32 log(2, x). Or, x = 32 log(2, x). So x / log(2,x) = 32. This can be solved by inspection, again beginning with powers of 2: x is divided by x as some power of 2, Trying a few values of x we see that log(2,256) = 8 and 256/8 = 32.

[@K.Klogic]

Thank you for your comment ❤️💖

[@rickkwitkoski1976]

@markharder3676 Thank you.

[@diwakaranvalangaimanmani3777]

This method for people who don’t know logarithm

[@b213videoz]

Very convoluted

[@antoniopetrolagarrigos9771]

​@@diwakaranvalangaimanmani3777if you know exponents you have to know logaritms😂

[@77das]

Сложно назвать это математическим методом решения, скорее простой подбор.

[@elmanmusaiev7003]

Это не совсем подбор. Методология и тактика тут есть. Подбор был сделан под конец, что подпортило эффект. Подбора можно было и не делать. Но понадобилась бы вычислит машина. И снова была б критика ))

[@ArinaAnn]

​@@elmanmusaiev7003 зачем машина, выше правильно про логарифмы написали. А показаний способ это все же подбор.

[@D0minate0]

​@@elmanmusaiev7003других вариантов решения нет?

[@elenaokon2736]

Curves 2^x and x^32 intersect twice: x at one intersection is positive, and at the second intersection is negative. So there should be two solutions.

[@K.Klogic]

Good dear ❤️

[@ПлатоновРоман]

Indeed

[@hkleiser5848]

i think you should show that the problem has only one solution to get all points, without that solution isn't full

[@Limited_Light]

It may be worth noting that 2 divides 2^x if x is natural number, so 2 would have to divide the right hand side as well. I think it's Euclid's Lemma that would then tell us x has to be at least a multiple of 2, if not a power of 2.

[@K.Klogic]

Thank you for your comment ❤️

[@Sci-Marvels]

Great observation! Indeed, if x is a natural number, 2 divides 2^x, so the right-hand side must also be divisible by 2. Euclid's Lemma is a helpful tool here, suggesting that x should be at least a multiple of 2, or even a power of 2. Thanks for bringing this up!

[@Limited_Light]

@@Sci-Marvels In hindsight, I wonder if there's a way to show x must be a *power* of 2. Overall, though, I think this and similar questions are related to the Lambert W Function, without getting into that.

[@Sci-Marvels]

@@Limited_Light that's a great question! It would be interesting to explore if x must be a power of 2. You're right that this and similar questions often relate to the Lambert W Function, even if we don't delve into that here. Thanks for bringing this up!

[@pub7496]

Thanks for the interesting problem. I liked the way you present. Subscribing

[@K.Klogic]

Thank u dear sir❤️💖

[@JSSTyger]

It looks to me as if there must be a solution between x=1 and x=2

[@K.Klogic]

Ok

[@rubixpuzzlechamp]

@@K.Klogicand a negative solution. There are 3 real solutions.

[@Swamisonic18]

How to find all solutions? Tell the method please🙏​@@rubixpuzzlechamp

[@chrisj5443]

A couple minutes of trial and error yielded x=1.0225 or thereabouts.

[@K.Klogic]

Thank you for your comment ❤️

[@Sci-Marvels]

Nice work! Sometimes a bit of trial and error can quickly lead you to a close approximation

[@داودزرابی]

Why 1:27 , was 1/32 * 2 ?

[@aleksandrajones4101]

So what about x=1.022 and x=-0.979?

[@luzak2k]

Error

[@daniel-mircea]

these are the other two real solution of the equation.

[@leviansilva1102]

A matemática é um jogo fascinante!!! Gostei.

[@K.Klogic]

Thank u for your comment ❤️

[@Alex_the_engineer737]

Если расписать левую и правую часть как функции, то за счёт чётности, функция f(x)=X^32 будет пересекать функцию g(x)=2^x в двух точках: одно решение потеряно. P. S. ради интереса забил в Desmosе эти 2 функции и абсциссы их пересечений равны 1.022 и -0.979. Вот так вот интересно.

[@K.Klogic]

Good work

[@ЛарисаРашевская-э3щ]

Думаю, что не ординаты, а абсциссы, но их три, еще 256

[@Alex_the_engineer737]

@@ЛарисаРашевская-э3щ Спасибо, исправил

[@Swamisonic18]

​@@Alex_the_engineer737 Yes same bro, but i can't find x= 256 in desmos plotting, why so? It shows only 2 intersections😢😢

[@Alex_the_engineer737]

@@Swamisonic18 Most likely, this is due to the fact that the polynomial of the nth degree has at most n roots. In our case, this is a polynomial of the 32nd degree on the right side, which essentially limits the number of roots of this equation. That is, in this equation it has already been proved that there are at least 3 real roots, and all the others are complex. (Unless we find other real roots)

[@jamesmarshall7756]

I use the Lambert W function to resolve x^32=2^x. «ln» both sides : 32 lnx= x ln2. So lnx/x = ln2/32 the same as (x^-1) lnx = ln2/32. x^-1 = e(ln(x^-1)) and lnx = -ln(x^-1). So we have : e(lnx^-1) x ln(x^-1)= -ln2/32. So lnx^-1 = W(-ln2/32). «e» both sides : x = 1/(e(W(-ln2/32))

[@K.Klogic]

Thank u for your solution ❤️💕💖

[@ЛарисаРашевская-э3щ]

Если условие найти натуральный корень, то решение класс, если количество корней, то по графику видно два и третий, который 256, его надо догадаться, за интересное уравнение спасибо!

[@K.Klogic]

Thanks 💕

[@Sci-Marvels]

Согласен, если условие - найти натуральный корень, то решение отлично подходит. Но по графику действительно видно два корня, а третий, который равен 256, нужно догадаться. Спасибо за интересное замечание и внимание к уравнению!

[@БогданМартынов-ж2у]

What about the second root in [-1, 0]? It's not a full solution, though, logic of solving is interesting

[@Observer1973]

You have lost two additional roots.

[@armandomeza6954]

Con logaritmo sale más rápido

[@ВладимирАгеев-з6ю]

in Russia we call it «The Monte Carlo method» or “sort through method”.

[@anishkumarmajumdar1322]

What method follow in the last step. Which rele used ?

[@jamesmarshall7756]

The difference between \(2^{1.0224}\) and \((1.0224)^{32}\) is approximately \(-0.00044\). This small discrepancy indicates that \(1.0224\) is a very close approximation to the solution of the equation \(2^x = x^{32}\). ChatGPT

[@K.Klogic]

Thank you for your comment ❤️

[@malinauagar4344]

This type of problem is still big problem fot gpt but it can solve it in the end, wiil wait fifth gen.

[@fani_uye]

Why the number keep getting bigger? N why equation is getting more complicated?

[@soshakobyan3123]

This is guess and check method, so you must prove there are no other solutions.

[@Durga.Devi123]

Great video, Sir how can we communicate with you regarding to any misunderstanding sie

[@K.Klogic]

How can I help you?

[@tihanyilaci]

Wow. So X^2=2^2 means x=2 is the only solution?

[@flap2380]

Before doing the first operation, you have to demonstrate that x =/= 0

[@aneedfortheory]

If you need to resort to manipulating in the way that you did, I think switching to logarithms is far easier. You simply have to find a multiple of 32 that includes as a factor a log to the base of 2 that generates itself....namely 256, which is 32 times log to the base 2 of, you guessed it, 256. If the kids are in the Olympiad selection process, that kind of arithmetic via logs is child's play to them.

[@ИванСафронов-ж9с]

А разве данное уравнение имеет не 3 решения? Автор указал всего 1, но их 3

[@Mb-logic]

Nice Explaination Dear.❤

[@K.Klogic]

Thank you for your comment ❤️💖

[@Mb-logic]

@@K.Klogic my pleasure ☺️

[@anurupdas9941]

By taking log this will provide trancedental equations to be solved graphically

[@K.Klogic]

Good

[@igogo1012]

Прежде чем использовать подбор следует сперва анализом доказать, что значение х хотя бы целое. Или, что оно не лежит в каком то диапазоне значений. А если хотите просто подобрать значение, то быстрее посмотреть ответ в интернете.

[@louf7178]

There's an identity in there somewhere, such as: for a^x=x^b and a^n*b; x=b*n

[@K.Klogic]

Good

[@nenemtiaof5016]

Where are the same bases ?

[@SameerMalik-ur8gb]

When u know the answer and try to get it anyhow 😂😂

[@KipIngram]

2^x = x^32 ln(2^x) = ln(x^32) x*ln(2) = 32*ln(x) x = (32/ln(2))*ln(x) x = e^((ln(2)/32)*x) A couple of iterations of the last one there yields x = 1.02239294025, which appears to be correct. Begin with x = 33/32, which is an easy to derive approximation (substitute x = 1+y and make the obvious approximations).

[@K.Klogic]

👍

[@alexeyevgenevich9821]

2^256 = 1,1579*10^(77)

[@mollockofmoon]

Using a graphic calculator, I checked the points of contact between the two functions other than x=256. After that, based on the two coordinate values, I was able to plug more specific numbers into the given equation and found two values ​​that were close to the correct answer. x=-0.979016935 x=1.0223929402

[@Lincoln-ht2dv]

Too many steps. Waste of time.

[@robertbecker959]

So how did you do it then?

[@lreactor]

What about the other two solutions

[@luxextenebrisss]

В 32 года узнала больше с ютуба, чем в школе

[@K.Klogic]

Good

[@Haloperidol-7]

Решается переобозначением x = 2^y, а затем простым подбором.

[@K.Klogic]

Thank u dear for comment 💕

[@АлексейЯкушин-щ4й]

But you don't proof that is only one solution. In naturals, yes, one 256, but in reals is one more root at interval [1,2]

[@anton-ke4qz]

But Why we don’t do any not equivalent transformation? Or do? To loose another solution in real numbers? Or there is negative solutions have been lost?

[@АлексейЯкушин-щ4й]

​@@anton-ke4qzThis solution is not correct

[@richardrakamazi5754]

x^(1/x) is a monotone decreasing function if x>e, then the equation x^(1/x)=2^(1/32) has at most one real solution. x=256 works, so no other real soultion.

[@daniel-mircea]

not at all. there are other two real solutions: x=1.022 and x=-0.979

[@richardrakamazi5754]

@@daniel-mircea ,,if x>e no other real solution than x=256

[@EliabBezerrademello]

I DONT UNSESTAND THE FINAL

[@jewi71]

But there is a second solution: X = 1,0223929... How can I find this second solution?

[@-Skynet-]

Это неверное решение

[@alexanderfishfinder]

@@-Skynet- Постройте графики функций y=2^x и y=x^32 и посмотрите где они пересекаются. x≈−0.97901693,1.02239294, 256

[@alexanderfishfinder]

32-довольно большая степень, чтобы на графике рассмотреть точку пересечения при x=256, поэтому для начала посмотрите на пересечения графиков y=2^x и y=x^2

[@stewardess-d4k]

@@-Skynet- Проверку проходит? 🙂

[@Sabri1729]

maybe you can say x=2^k (where x>0) and get 2^k=32.k or f(k)=2^k-32k anf f'=lnk.2^k-32 than pick a root, lets say k=1 and apply newton-raphson algorithm, probably in a few steps you will get a very close numerical solution. for negative root you can say x=-2^k (x

[@glitch9643]

Cant we use log here?

[@K.Klogic]

Try first

[@WiXGaminG]

Your mind has to work in a strange way to be able to comprehend such things :D to me its pure magic😂

[@FarazBeg]

I don't mind the method, but do mind the time it took

[@kusumikasiddhanta607]

Using logarithmic fuction is the true solution for this particular sum...because it is not a solution unless you are able to solve all the similar types of sums using that method, irrespective of the base and the exponent...for example if it was 31 instead of 32...using this method would have fetched us nothing at all.

[@radutrimbitas4930]

there are 2 more solutions, x\approx -1.022 x\approx -0.979

[@jeanpierre7971]

Don't you need to prove that x -> x^(1/x) is bijective?

[@Drestanto]

not really as long as you do the verification at the end

[@jeanpierre7971]

@@Drestanto Then what guarantees that there is no other solution?

[@Drestanto]

@@jeanpierre7971 yes, there's no guarantee for a unique solution If it's bijective, it has a unique sol What I meant is that x -> x^(1/x) does not need to be bijective to make one solution works as long as there is a verification at the end

[@jakefromstatefarm6969]

Given that this is a competition question posed to us, there is no need to prove anything. All that is needed is determining a solution, not proving that it is a unique solution.

[@pedjolinko]

@@jakefromstatefarm6969 You are very wrong. First of all, there are two solutions though only one is an integer.

[@Songfugel]

Pretty nice format, was easy to follow even without commentary

[@K.Klogic]

Thank you for your comment ❤️

[@АнтонСергеев-ъ9н]

Дайте угадаю: через логорифмы?

[@vishnusir7838]

Log kis din Kam ayega Bhai

[@EasyMathSteps-SolveItEasy]

Pretty elegant solution! Thank you!

[@K.Klogic]

Thanks for your helpful comment ❤️

[@Durga.Devi123]

Sir if there is any G chat available? Through gmail so can discuss further regarding to topic

[@etcesr]

It is like a endless math nightmare 😄 but thanks 👍

[@laurentblanchet1803]

Ne faudrait il pas par la suite prouver l'unicité de la solution??

[@MrTaiado]

há mais de uma maneira de fazer, mas foi uma bela solução!!!

[@K.Klogic]

Thank you for your comment ❤️

[@СергейПопов-э9о]

Thanks for the video. Maybe someone knows how to find these two more solutions for this problem? (~ -0.979 and 1.0225)

[@K.Klogic]

Thank you for your comment ❤️

[@screemer949]

Yes, I published actual solution in comments

[@荒巻-b8m]

Another answer is approximately x=1.023.

[@K.Klogic]

Good work

[@НашалабайРаджик]

Повезло, что показатель степени справа не 1024 😂

[@барченкоша]

Шок задание с выпускного экзамена гарварда 2^x=4, если x={R}

[@蔡木章]

Very good thanks

[@K.Klogic]

Thanks for your comment ❤️💖

[@cosmosapien597]

Why do you write like that?

[@EliabBezerrademello]

I DONT UNDERSTAND THE FINAL

[@K.Klogic]

Watch again sir💕

[@hectoraular6180]

En ese caso no era necesario decir? que X diferente de 0 y X > 0

[@agniksen8507]

Will the person even have the time left to finish the exam?? 😒😒

[@K.Klogic]

No dear sir ❤️💖

[@recepgurkanhosman155]

Cevap 1 oluyor 2 üzeri 5 x in üzeri 32 den x le 2uzeri 5 yer değiştirme yaparak üste 5 üzeri x kalır oda x sıfır dan 2 ye eşitlenir.

[@DumAndSmart]

Take log two of both sides then you get X equals log base two of x multiplied by 32 other words what number when you take it’s logarithm it does the same thing as dividing it by 32. You can make educated guesses until you stubble upon 256.

[@Sci-Marvels]

Great approach! Taking the log base 2 of both sides definitely simplifies the equation to x = log base 2 of x multiplied by 32. As you mentioned, making educated guesses can eventually lead to finding that x = 256. Thanks for sharing this insightful method!

[@vasiliynikiforov1976]

Икс ведь может быть ноль?

[@АлексейЛуз-б3й]

Но x=256 не является корнем уравнения! Компьютерные вычисления это показывают. Примитивный скрипт на питоне. Если строить график, то будет два действительных корня.

[@K.Klogic]

May be u r right dear❤️

[@sarvarmahmudov6276]

X=32 can be a root too

[@PawełPiotrWaz]

This world is not normal.

[@ЭдПетр]

-0.979 approximately next solution. There are three solutions

[@K.Klogic]

Thank you for your comment ❤️💖

[@mcrow312166]

I love the way you communicate without words.

[@K.Klogic]

Thank you for your comment ❤️

[@davidbrown8763]

...and with decent music, unlike so many other channels. I have just liked and subscribed.

[@K.Klogic]

@@davidbrown8763 thank you dear sir.supor my channel thank u💖❤️🎉

[@davidbrown8763]

@@K.Klogic I am a supporter.

[@K.Klogic]

@@davidbrown8763 thank you dear..

[@tshepomalalane4140]

Beautiful 🫡

[@MrPsubbara]

This equation has infinite answers. That's only one of the answers or roots

[@YasinBEY-eo1ng]

Şöyle bi bakayım dedim denerken buldumn😅256 deneme yanılma cikiyor

[@stevenschmidt]

256 isn't the only solution.

[@K.Klogic]

Dear sir u can solve ❤️💖💕

[@jojojo7333]

Now solve 4.67^x=X^23.4

[@erkincakar7966]

2 base 4=4 base 2 32×4=128×2=256

[@DuyTran-uj1cv]

why x 1/x

[@SHIVA7278]

Exceptional

[@lool9935]

Bro this not a standard analytical way

[@K.Klogic]

Ok

[@wwxyz7570]

WolframAlpha gave me 256 in less 1 second

[@K.Klogic]

Great ❤️💖💕

[@aranarus]

The author does not know mathematics. This problem has 3 solutions, and he showed only one. It turns out that the problem was not solved.

[@busraakagunduz.]

Soruyu gereksiz uzatmış. Her iki tarafın logaritmasını alsa zaten sonuç çıkardı

[@ozencgencmert]

I solved this question using logarithms in 1 minute.

[@K.Klogic]

Good work dear❤️💖

[@DumAndSmart]

256,1.022,-0.979

[@K.Klogic]

Thank you for your comment ❤️💖

[@Yacine-e9p]

Suite de fonction ; introduction à la série de Fourier

[@yogamulyadi2046]

x=2^n=32n n=2^(n-5)=8 x=2^8=256

[@TheOnlyAnonymousMan]

Lovely

[@K.Klogic]

Thank u dear for your comment ❤️💖

[@tahamuhammad5962]

TZZ {4,2,1} Tzz is the Taha + Collatz brand. Factories of T-shirt, sports shoes, engineering machinery and tools, airplanes, cars, trains and ships must use this mighty sign that signifies victory and the solution of Collatz. So factories must agree with me to use this mark. Author: Taha M. Muhammad/ USA Kurd Iraq Bakuage Thank you prepare your prize

[@ardalex4277]

The answer is not complete. I guess there is another solution in (-1,0) interval. And another one in (1, 2) interval.

[@K.Klogic]

Yes dear i mention already

[@laith.alzubaidi]

using log is simplfy the equation