A beautiful math question for advanced students

[@justliberty4072]

I'm 63. I have a PhD in engineering. I've never heard of Lambert's W function.

[@assiya3023]

العلوم في تقدم مستمر ، يجب علينا أن تحيين معلوماتنا دائما، ملحوظة أنا أيضا أبلغ من العمر 58 سنة ولدي دكتوراه في الهندسة الكهربائية . ولازلت أتعلم

[@USCsteveO]

​@@assiya3023عمري 40 عامًا تقريبًا وأنا⚡ مهندس برمجيات. التعلم هو الطريق.

[@donmoore7785]

I hadn't either, BS in engineering in '83.

[@SebastianMedina]

Me too. Electronic Engineer '07

[@AfricaWorldTV411]

Same here 42 years old,but it's always good to learn

[@rorydaulton6858]

One thing that I did not hear you explain is that the Lambert W function has countably infinite many branches in the complex numbers, so there are many complex solutions. You gave only one of them. Another is approximately 0.06396 - 1.0908i, yet another is 1.2484 - 5.5045i, and so on.

[@onradioactivewaves]

...and so 4ᵗʰ.

[@papkenhartunian186]

How do you find complex roots?

[@rorydaulton6858]

@@papkenhartunian186 (My previous reply seems to have disappeared.) I used the SciPy package with the Python programming language. Search for Scipy and Python for details. I do not seem to be able to paste the relevant URL here. The WolframAlpha web site can also calculate values of the Lambert W function: just use the name "productlog" for the function. I have not yet figured out how to get WolframAlpha to choose a branch of that function other than the main branch.

[@BSnicks]

Take it easy on him. I don't think he understands what the Omega function is yet.

[@onradioactivewaves]

@@papkenhartunian186 research the Lambert W function, that is the solution.

[@StatisticalLearner]

Well, beauty is in the eyes of beholder. So there is no sense of arguing whether this solution is beautiful or elegant. It however, borrows something itself is quite esoteric and difficult to understand, vis a vie, the Lambert W functions, to seemingly solve another enigmatic equation. Except it actually does not solve it in the close form (that can be expressed with elementary equations), because Lambert W function can only be solved numerically. If that is the case, it is much easier to solve the original equation numerically. In addition, the solution in this video actually mis represented the value of Lambert W function. It is a function that has many (complex) branches leading to many roots, and the true beauty is in how these branches, and roots are related to each other. For those looking for beauty in mathematics, you are more likely finding it in the Lambert W function itself. It is also worth noting, the Lambert W function is actually fully developed by Euler, and it has an important place in Dirac's quantum mechanical representation of chemical bonds and other important physical phenomena. And the fact that Euler developed the function and derived its properties a couple hundred of years before its application in science is mind-blowingly beautiful.

[@javgrzg]

You nailed it mate

[@shashwatraj2381]

Tldr?

[@billwindsor4224]

@StatisticalLearner I agree with @javgrzg, excellent elaboration on the Lambert W function and Euler’s and Dirac’s work, thank you.

[@billwindsor4224]

@@shashwatraj2381 As you progress in math, you will find it necessary to read and compute complex material. Check out the works of Srinivasan Ramanujan.

[@PavelKucera-bd9dd]

Sorry, but 4^(-0.0887+1.512i)=-0.4434+0.765i. Thus 4^x≠x. Your root is not a solution of the equation. The first root of this equation is 1.248-5.505i since 4^(1.248-5.505i)=1.248-5.505i.

[@leonheintzen3431]

je confirme !

[@papkenhartunian186]

How did you find your answer?

[@dantefitti6678]

Bravo!

[@jbangz2023]

4^(1.248-5.505i) = 1.24446 - 5.50222 i

[@LelekKozodoj69]

​@@papkenhartunian186 Newton's method. z(n+1) = z(n) - f(z(n))/f'(z(n))

[@Akenfelds1]

This video labours dead simple mathematic steps like it was high school mathematics, but offers no explanation of Lambert's W function. It's like watching somebody make a ham and cheese sandwich with a step-by-step guide as to how to slice ham but no explanation as to where bread or cheese comes from.

[@donmoore7785]

Not only that, but how one finds complex solutions involving the Lambert W function. He pulled that from somewhere.

[@svenbalzer6763]

Hello, It’s Only forn Part Highierer Like int Realschoolniveau Perhaps. Formytaste Is This Not Highschool or Studyniveau. Lg Sven BALZER

[@ChrisBoshuizen]

I came to say the same thing. But I'd say it's more like mindlessly rearranging the table place setting, and then pulling a fully made ham sandwich from the fridge. Very disappointing video.

[@yama8483]

There is not really a lot to explain about the Lambert W function that is actually useful for the purposes of the video. It is simply a function that, for any x that can be described as x= a*(e^a), W(x)=a. It is actually a multivalued function, due to the nature of complex logarithm, but most of the time people focus on the principal branch, and with the exception of one, all other branches only produce complex results. It is also sometimes called the product logarithm. It has applications in higher levels of physics, mathematics and statistics, but it is not really ever taught outside of university, and even then only in the fields that actually use it. Nobody calculates W(x) by hand; it cant be articulated using elementary functions and can only really be explained as the converse relation of f(x)=xe^x; and is almost always kept in notation in Mathematics. The only other thing that would actually be useful to elaborate is that it is, for the principal branch, only defined for x ≥ -1/e in the real numbers, if you go below that value the results are complex numbers, which does happen in the video.

[@Anti-You]

I was an applied math major and I've taken complex analysis. I've never heard of the Lambert W function. Wolfram classifies it as "Miscellaneous Special Functions" and it seems that it has this one useful quality. It'd be good if you'd have an example that doesn't result in an imaginary answer. I'd like to see it.

[@tuho2977]

No need to complicate the problem. From the equation x cannot be negative. With x>=0, taking ln of both sides and using the upper bound of a natural logarithm, we can easily prove that x*ln4 > lnx so there’s no solution.

[@CLOVERYAU]

No real solution but it has complex solution as explained in the video.

[@22tirthsavaliya36]

X= 0 is a possibile thing

[@LeoV007]

​@@22tirthsavaliya36No

[@matthewbertrand4139]

​@@22tirthsavaliya36it's not, just by inspection. you get 1 = 0.

[@22tirthsavaliya36]

@@matthewbertrand4139 yes I did something wrong

[@gamertz505]

Beside the math did anyone try to remove hair from screen like me in beginning?😂😂🤣

[@RadicalCaveman]

It's not a hair. It's a plutonium shaving.

[@gamertz505]

@@RadicalCaveman 😂🤣😂🤣😂🤣🤣

[@olmynuwen]

Somehow I do not feel enlightened

[@zdrastvutye]

this time it will work also for a=1.1 or a=2 or a=4, see line 10: 10 a=4:print "higher mathematics-a beautiful math question" 20 sw=.1:b=sw:goto 50 30 csc=b/a^b:if csc>1 then stop 40 c=acs(csc):snc=sqr(1-csc^2):dg=c/snc/a^b:dg=dg-1:return 50 gosub 30 60 b1=b:dg1=dg:b=b+sw:if b>100 then stop 70 b2=b:gosub 30:if dg1*dg>0 then 60 80 b=(b1+b2)/2:gosub 30:if dg1*dg>0 then b1=b else b2=b 90 if abs(dg)>1E-10 then 80 100 print b,c 110 print "exp(";b*ln(a);"+";ln(a)*c;"*i)=";b;"+";c;"*i or" 120 print a;"^(";b;"+";c;"*i)=";b;"+";c;"*i" higher mathematics-a beautiful math question 0.250501609 1.39285046 exp(0.347268968+1.93090073*i)=0.250501609+1.39285046*i or 4^(0.250501609+1.39285046*i)=0.250501609+1.39285046*i > run in bbc basic sdl and hit ctrl tab to copy from the results window. if there is a mistake, let me know. see also wolframalpha

[@PNLMaths]

I am a four year trained Maths teacher with a pure Maths degree and 35 year experience of teaching all levels from year 8 to year 12. From the graphs of y = x and y = 4 to the power of x, we can easily see that they have no intersection. That's enough to conclude the above equation has no real solution. That's it. What is Lambert function? Thanks.

[@Russ--R]

I'd stick the two functions in Excel and have a look at where, if at all, they cross.

[@VengerVideoGamer]

Exactly. I did my maths degree 30 years ago and never once came across the Lambert W function as it seems that it's not widely taught. Indeed, this is what Wikipedia has to say about its usage : In 1993, it was reported that the Lambert ⁠W function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges-a fundamental problem in physics. Prompted by this, Rob Corless and developers of the Maple computer algebra system realized that "the Lambert W function has been widely used in many fields, but because of differing notation and the absence of a standard name, awareness of the function was not as high as it should have been."

[@752castro]

The Lambert function is the inverse function of the function that multiplies a number X by its exponential e^X . This means that the Lambert function transforms the expression “X times the exponential of X” back into X . In other words, when you have a value K , the Lambert function, denoted as W(K) , solves the equation that states “X times the exponential of X is equal to K.” Therefore, when you calculate the Lambert function for a value K , you find the value of X that satisfies this equation.

[@RexxSchneider]

The part which was missing was any proof of the assertion (around 7:34) that the graphs of y=4^x and y=x have no intersection. It turns out that y=a^x and y=x have no intersection when a is larger than about 1.44, but there is an intersection for smaller values of a. If you've spent time looking at these sort of problems, you would know that y=4^x and y=x have no intersection, but to baldly assert it is a flaw in the presentation.

[@zdrastvutye]

i have changed that code a bit 10 a=4:print "higher mathematics-a beautiful math question" 90 c=acs(1/ln(a)): b=c/ln(a)/sin(c):print b,"%",c: 100 print "exp(";b*ln(a);"+";ln(a)*c;"*i)=";b;"+";c;"*i or" 110 print a;"^(";b;"+";c;"*i)=";b;"+";c;"*i" this question will become really difficult if a in line 10 is smaller than exp(1), so cos(e+f*i)=d (with d>1) cos(e)*cos(f*i)-sin(e)sin(f*i)=d and now there is a relationship between cos and hypcos as well as sin and hypsin via the imaginary number "i"!

[@winstongludovatz111]

This is painful, the most trivial inequalities will show that there is no real solution. Higher mathematics?

[@adrianlautenschlaeger8578]

I think it's not painful because of missing real solutions. The most equations/functions are really hard or impossible to invert. Popular example ist LambertW, the inverse funktion of xe^x Or try to invert a quintic function ;-)

[@paulo33]

this is not higher mat .

[@mathematicsquiz]

kzbin.info/www/bejne/mHmuk5udd6Z1jtk

[@tungyeeso3637]

As usual, I would prefer using the graph approach. Since the answers are approximates, a graph works a lot simpler. 😊

[@davidchilds9590]

You have explained an answer that does not exist. You have not explained how you found the complex solution.

[@KLONDYKE1111]

You speak too fast!

[@BSnicks]

LOL

[@ioshift]

The minimum value of lnx/x is 1/e

[@IITJEEAspirant2025-fj6hj]

4^x=x By logarithmic approach we have, x ln4 = lnx => ln4 = (lnx/x) 2×ln2=(lnx/x) => 2×0.693=(lnx/x) =>1.386=lnx/x By graphical approach lnx is an increasing function But x is a function which is symmetrical to two quadrants like y=x linear function type But the rise in y=x graph would be much more as compared to lnx The question itself asks for the point of intersection of both the curves ! Hope it helps!

[@春-i4z]

ax^2+bx+c=0 . For this quadratic equation, if the discriminant is b^2-4ac

[@IITJEEAspirant2025-fj6hj]

@@春-i4z I am saying the point of intersection of the curve y=x and y=lnx curve ik if d

[@春-i4z]

I've checked your ID using AI. Good luck with your entrance exam studies. . "IIT JEE Aspirant 2025" refers to students who are looking to get into the Indian Institutes of Technology (IITs) in 2025. To get into an IIT, you need to pass the Joint Entrance Examination (JEE), a highly competitive exam. JEE is divided into two main stages: JEE Main and JEE Advanced. . JEE Main is conducted by the National Testing Agency (NTA) of India and is the foundation exam for admission to engineering colleges and other technical universities across the country. JEE Advanced is the final exam for students who have passed JEE Main and are looking to get into an IIT. . Students who plan to take the exam in 2025 are often starting their preparations now, preparing for the exam through study plans, mock exams, and past exam answers.

[@ya_ne_beliy]

Lets suggest we have >0 real roots Also we can do: f'(x)=g'(x) where f(x), g(x) left and right parts of eq. We can get f'(x)=1 => x ~ (-1;0) So we can draw 3 funcrions: f(x), g(x) and h(x) where x = arg(f'(x)), f'(x) = 1 We can see g(x) >= h(x) > f(x). So g(x) > f(x) for every single x, but g(x) = f(x), contradiction. We have 0 real roots.

[@padraiggluck2980]

The graphs of the functions y=x and y=4^x do not intersect.

[@jstarks123]

Which is why the solutions are complex.

[@padraiggluck2980]

@@jstarks123 R2 is isomorphic to the complex plane ergo the graphs do not intersect there either. The only ‘solution’ is bogus.

[@spacelem]

​@@padraiggluck2980y=x²+1 and y=0 do not intersect in R² either, and yet we know that intersects at the points (±i,0).

[@padraiggluck2980]

@@spacelem x^2+1 has nothing to do with the given problem.

[@spacelem]

@@padraiggluck2980 except that it fits the same properties you gave for saying "the only 'solution' is bogus".

[@JacobWakem]

You can also solve this using an infinitary expression and finding the fixed point.

[@noomensaidani131]

Parachuted responses. What is w fonction, in which field you've applied ln on x...... Etc

[@ajitandyokothakur7191]

You keep repeating "Natural log Natural log..."). It is probably your verbose but some might find it confusing. Dr. Ajit Thakur (USA).

[@Andrew-II]

I found a solution in parametric form: x = a + i * sqrt(4^(2*a) - a^2), where a is any real number. I did not use the Lambert-W function, only the Euler transformations, under the assumption that x is a complex number of the form a+i*b. It really works, I've checked numerically.

[@2Eさん]

I got an alternate solution of W(-ln4)/-ln4 e^(xln4)=x (ln4)e^(xln4)=xln4 (ln4)=xln4*e^(-xln4) -(ln4)=-xln4*e^(-xln4) W(-ln4)=-xln4 x= W(-ln4)/-ln4 This should hold for complex X no?

[@jcolinmizia9161]

4^1 = 4 4^0 = 1 The error in the problem comes from recognizing that the cannot be greater than zero, as the value will always be greater than the exponent. But also that it can’t be less than zero, as the right hand side cannot be negative. So there is no real number solution to this problem.

[@papkenhartunian186]

Could you explain how you found the complex root?

[@urban_sculptor]

G**gle Scipy lambert. There's a good explanation and example how to find complex roots

[@zdrastvutye]

i just have applied de moivres formula which is based on the power sum production of euler's number 2.71828: e^(i*x)=cos(x)+i*sin(x) so exp(ln4*(b+c*i))=b+c*i: 10 a=4:print "higher mathematics-a beautiful math question" 20 sw=.1:c=sw:goto 40 30 dgu1=sin(c):dgu2=ln(a)*sin(c)*cos(c):dg=dgu1-dgu2:return 40 gosub 30 50 dg1=dg:c1=c:c=c+sw:if c>100 then stop 60 c2=c:gosub 30:if dg1*dg>0 then 50 70 c=(c1+c2)/2:gosub 30:if dg1*dg>0 then c1=c else c2=c 80 if abs(dg)>1E-10 then 70 90 b=c/ln(a)/sin(c):print b,"%",c:print "exp(";ln(a)*b;"+";ln(a)*c;"*i)=";b;"+";c;"*i" higher mathematics-a beautiful math question higher mathematics-a beautiful math question 0.796835729 % 0.765050302 exp(1.10464888+1.06058492*i)=0.796835729+0.765050302*i > run in bbc basic sdl and hit ctrl tab to copy from the results window. however,i avoided the "tan" function which is not continuous.

[@LelekKozodoj69]

You made a mistake. -0.0887+1.5122i is a value of W(-ln(4)), and not 1/exp(W(-ln(4)). Therefore it's not a value of 'x'. The solution is x = 1/exp(-0.0887+1.5122i).

[@SylvainKnowsIT]

Nice reminder. But could you point me to some material explaining the W function?

[@LelekKozodoj69]

Wikipedia. Lambert W function.

[@SylvainKnowsIT]

Thanks, ​@@LelekKozodoj69

[@mdrokebtamim648]

This question itself wrong. How it is possible to X is more bigger than X 😂😂😂. X=4^x impossible.

[@antennaist]

Exactly.

[@shipsahoy1793]

when mathematics becomes impractical, it essentially becomes a useless waste of time .

[@peterreali3950]

@@shipsahoy1793 x = -1/2 is the simple solution -1/2 = 4^-1/2 = = or - 1/sqrt(4) = + or - 1/2 choose negative root so x = -1/2

[@student6140]

Not if x is 0

[@shipsahoy1793]

@@student6140 if zero is the only solution, then what's the point of the equation, other than noting a specific math property ? ..Mathematics consists of properties, identities, rules, and procedures, etc. but exists for the purpose of actually quantifying real world phenomenon, or solving math equations that get you there. There really is no point to an equation like the one written here, other than to say any integer to the power of zero equals zero.

[@ptrblz]

Nice, now if you rotate the linear function a bit you might get one or two real results, I was interested whether I can find the tangetial one and it seems I did for equation 4^x=e*ln(4)*x there shouod be just one real solution

[@luckyluk4]

you found a complexe root using a complexe way😅 ! there a simple way just from line 4 . you plot the function ln (x)/x - ln4 = 0. you ´ll find the graph below and not touching the x axis, which means no real roots .

[@spacelem]

"natural log natural log 4" is ln(ln(4)). This would be very difficult to follow along with just the audio. Just say "log 4", we know it's base e.

[@fernandoangulo1960]

Hello, According to the fundamental theorem of algebra, a polynomial of the fourth degree must have four roots, including real and complex roots. Where are the other three roots?

[@imperialblast]

Theres no polynomial of degree 4 here (notice that it’s in the form a•b^x, whereas for polynomials all terms are in the form an•x^n)

[@yogesh193001]

I feel Lambert W is just a cheat😂😂

[@bookashkin]

Sure, but so is the square root function :)

[@packerfan2010]

so are the complex numbers lol. They were literally invented to solve the equation x^2 = -1

[@dr.rolandzagler8831]

Hello, this is a nice equation with a expectable complex solution. I enjoy it. Nevertheless plz can you proof your complex value: Wolfram Alpha gives 0.06346 - 1.09084 i. The same value I obtain by calculating x = -W(-ln4)/(ln4) in solving your eq. with a second method (Initial deviding the eq. by 4^x.) Many greetings - Roland

[@zunden2]

Pertanyaan ini tidak ada hasil jawaban, meskipun x sebelah kiri bernilai 0, x di sebelah kanan bernilai 1, tapi bila x sebelah kiri bernilai -1/2 maka x sebelah kanan bernilai 1/2.

[@nooruddinbaqual7869]

Lambert W function is confusing Is it of any practical use or serves as brain teaser only?

[@bookashkin]

W is defined as a compositional inverse to a more familiar function. Shorthand (so yes, practical). For example sqrt(x) is shorthand for a number a such that a^2=x. Arctan(x) is shorthand for a number a such that tan(a)=x. Ln(x) is shorthand for a number a such that exp(a)=x. W(x) is shorthand for a number a such that a*exp(a)=x.

[@TonyDig100]

infinitely many complex solutions -productLog(n, -log(4))/log(4) where n is an integer

[@franzxavereiholzer8499]

The numeric solution he gives is wrong

[@АртемХарченко-й3б]

The solution can be easier determined by grafic

[@skateordiee]

Wait, I think I figured it out… X = X lmao 😂

[@GuoweiMa-k9t]

Every time i can't get into sleep then i choose ur video😮

[@xypherdrakeinsignia21]

You can hav 4 as 2^2(x) = x^1 The solution of the x power can be solved = 1/2 Substitute the x power to 1/2 for 4 or square root of 4 is 2 hence the variable x is equivalent to 2

[@achomik1999]

so u input x=1/2 and output x=2? lol this isn't c++

[@kendrickainsworth8376]

my initial thought was : take Fourier Transform of both side and reverse it

[@AlexandruHusti]

4^x = f(x); geometricaly will be one exponential x = f(x); geometrical will be one line this two function cross between them in 0,3

[@AlexandruHusti]

when they acrossing should be the result

[@allihhussainmazari168r]

Why don't you use differentiation dy by dx or integration rather than going this long route

[@janwbo]

There is an inaccuracy in the video at 7:03. Instead of “x=” it should say: “W(-ln(4) =…” Using the value given in the video x can then be determined to x=e^-W(-ln(4)) \approx 0.06396-i*1.09084 This value should be given as the answer. In order to double check enter this to Wolfram Alpha: 4^(0.06396-i*1.09084)

[@verdergreen32]

Such a task will never appear in reality and such an answer is of no use to anyone, because what to do with it?

[@Kanal263]

Es gibt keine reelle Lösung. Das wird durch eine Kurvendiskussion von f(x) = 4^x - x deutlich.

[@noname-ed2un]

Can you please explain this statement. I don't understand

[@ffggddss]

Even by just imagining the graphs of y = 4ˣ and y = x, you should be able to see that they don't intersect, and therefore, there can be no real solutions to 4ˣ = x Any solutions must be complex, non-real. I suppose you could let x = re^(iθ) = r cosθ + ir sinθ or just x = u + iv, and take it from there. Looks like this is heading into Lambert-W function territory, which in complex space can get complicated. Let's see what the video does with it . . . Fred

[@Hellspawn100]

Never heard of the Lambert function as my peers. I'd probably go with a graphical method with the assumption that x € R. And if I go down that path this equation has no real solution.

[@betacentauri6830]

Die Antwort ist einfach. Es ist x = 4^x. Denn oben drüber steht ja 4^x = x. Aber Scherz beiseite, das thumbnail erweckt den Eindruck, es gäbe eine reelle Lösung. Aber da ist auch Einstein abgebildet, von daher könnte man denken, es geht um Relativitätstheorie mit Tensorrechnung. Dann wäre auch x = 4 eine zulässige Lösung, denn 4^x wäre dann nicht 4 hoch x, sondern nur 4 Längeneinheiten in Richtung x. Also 4 = x.

[@nayeemsschool1653]

haven't you studied Cambridge syllabus? simply use iteration method. its done

[@jim2376]

Mr. Lambert enters the room.

[@semihdemic6448]

There is not any question on the page. There is only an equation. It can ask what X is.

[@giovanniviglietta731]

This is NOT a math olympiad question. This is the opposite of a math olympiad question.

[@leolacic9442]

ha ha ha ha ha ha

[@giovanniviglietta731]

@@leolacic9442 What's funny?

[@ghostsofbeauty.9346]

of two poss. answers i find here. 1 is. 4x. 2 is. 4x =x x= ?..

[@areibielghoul3150]

To solve the equation \( 4^x = x \), you can use numerical methods or graphical methods, as it doesn't have a simple algebraic solution. However, we can approximate the solutions as follows: 1. **Graphical Method**: Plot the functions \( y = 4^x \) and \( y = x \) on the same graph and find their points of intersection. 2. **Numerical Method**: Use numerical approximation techniques, such as the Newton-Raphson method. By inspection or using numerical methods, we find that: - One solution is \( x \approx 0.64 \). - Another solution is \( x = 2 \). So, the approximate solutions are \( x \approx 0.64 \) and \( x = 2 \).

[@n00bxl71]

Intersection of y=4^x and y=x on a graph? What universe are you living in? They clearly never touch.

[@donmoore7785]

You are mistaken. There is no real solution, and the graphs do not intersect.

[@ГосударственныйДепартамент]

mathematics: no room for error higher mathematics: *excist*

[@jeannieheard1465]

Was there ever a mathematician with a sense of humor? OK, Einstein.

[@YassJ-jd5sz]

No ones knows about this W function. I think it would have more interesting to ask to prove that there is a unique solution instead.

[@kopi314]

Why ln and not log2 or log4 is used.

[@AI_Look_Girl_2023]

The result I calculated using the AI ​​program is: around X≈0.05

[@Gnowop3]

Why make it so complicated if it is required to be in the Real Domain. Just reason from the two curves of y=4^x and y=X and they dont intersect and hence no real soultion

[@sergeisychev6824]

Let’s calculate and verify if 𝑥 = - 0.0887 + 1.512 𝑖 is indeed a solution to 4^x=x -- The evaluation of the equation 4^x=x with x= - 0.0887 + 1.512 𝑖 yields: Left side (4^x): - 0.4434+0.7651 𝑖 Right side (x): - 0.0887 + 1.512 𝑖 Indeed, it’s not a solution. This verification suggests that the complex value proposed does not satisfy the equation under standard complex arithmetic. -- Moreover, I can give an analytical proof that this equation has no solutions: separately for the entire set of real numbers and separately for complex ones.

[@LloydCash-he1qv]

The correct answers are, as explained in the video, x = 1/exp(W(n, -ln(4))) where n is an integer giving x ~ 0.06396 + 1.09084i with n = 0 One gets other answers depending on the value of the integer n. Alternative correct solutions also are x = -W(n, -ln(4))/ln(4) NONE OF THESE INCLUDE THE AUTHOR'S ANSWER AT ALL! x ~ -0.0887 + 1.512i ISN'T A SOLUTION TO THE QUESTION IN ANY SHAPE OR FORM!!!!

[@user-gi2uh2vc6k]

Just draw both functions... Do they intersect each other? No. Ergo: no solution

[@user-gi2uh2vc6k]

Fun fact, (1/4)^x = x would have a solution though, and that's 0.5

[@desmondaubery8446]

If you try plotting y=4^x - x, there are no roots in the real plane.

[@春-i4z]

Instead, there are an infinite number of solutions on the complex plane.

[@hefazlala]

I have solved this one, sometime ago very easily. This does not belong to higher mathematics.

[@marcusray1638]

4 = x/logx ,the standard decreasing function for all x greater zero. by graph or try and error. it is easy to get the answer easily 😂!

[@question1answer1Q1A1]

2^x = x , if we put x = 0 + 1i then 0 + 1i ≈ 1 ,

[@makehimobsessedwithyou6412]

Why not using log

[@noname-ed2un]

I thought about that at first. But after trying to solve it I couldn't find an answer

[@JyotikaSharma-w5v]

I think the answer is x=x/2 log 2 Or x=x/log 4.

[@wongkienloon]

Balance Method and Blind Method

[@tmjcbs]

I'm eagerly awaiting the video of 5^x=x...

[@春-i4z]

Hello all 5^x=x, in this case, x(n)=a+bi, n=1~10. x(n)：a+bi x(1)：-0.0107508384725557 -0.982787226228599i x(2)：0.981550237762412 -4.75343010959909i x(3)：1.3507162537896 -8.68808537617985i x(4)：1.57962629620438 -12.6104510704671i x(5)：1.74626963395653 -16.5264299576489i x(6)：1.87745083605957 -20.4388891716296i x(7)：1.98567056750388 -24.3492080241234i x(8)：2.07779185002501 -28.2581242813096i x(9)：2.15799411827267 -32.1660684271127i x(10)：2.22901340052689 -36.0733091326578i That's all. Thank you.

[@春-i4z]

Hello all Verification： 5^(-0.0107508384725557 -0.982787226228599i )=-0.0107508384725556... -0.982787226228599... i That's all. Thank you.

[@bigscrounger]

Is there a reason why you keep saying 'natural log natural log 4'. It's just supposed to be said once, unless you're applying the function twice...

[@TristanBBBBB]

Yeah. That was weird.

[@spacelem]

I kept hearing that too, came to the comments to mention it. Personally I'd just skip the "natural" part after the first time and just say "log x", unless we change base.

[@DarVV]

Some tree logs laying in a park or forest are an example of the nature 😊 All of you have right about it. We should simplify our speech. Math itself can be complicated on academic level. Sometimes some blind people can hear this video too.

[@jakemccoy]

@@spacelemMost students learn that “log x” is assumed to have base 10. It would not make sense to clear up the language only to cause potential confusion elsewhere.

[@spacelem]

@@jakemccoy do they? I have a maths degree and I don't remember at any point log being assumed to be base 10. Even at school.

[@williamspostoronnim9845]

Стоило бы остановиться на том, как вычисляются комплексные значения функции Ламберта.

[@simokhaoua9496]

x=-0.0887+1.512i is not a solution 4^-0.0887+1.512i = −0.4741+0.7472i Clearly, 4^x ≠ x since the values do not match. The values differ considerably in both real and imaginary parts.

[@春-i4z]

This is a solution for W(-ln(4)), but not for x. The value we are looking for is X = 1/e^(W(-ln(4))). W(-ln(4))=-0.0887+1.512i Therefore, X = 1/e^(W(-ln(4))) =1/e^(0.0887+1.512i)=0.06396 -1.0908i This value is one of the solutions the author should be looking for. Therefore, the following relationship holds: 4^(0.06396 -1.0908i)=0.06396 -1.0908i. That's all. Thank you.

[@undecidable]

You take time to explain the most rudimentary algebraic steps, but fly over the use of W. Come on...

[@donmoore7785]

Yes he explained middle school math steps but brushed over Lambert W and in particular finding a complex solution.

[@musabsaleem3611]

Can you solve this equation: 9x square + 21x -8=0

[@florinnegretu7425]

4^x>x whatever x....asta la vista, baby!!!solving in R, of course...

[@halloussi]

Lambert's W function : we have done it in Math Spe ... an advanced function

[@cykkm]

W_k(x) is branch-cutting at every integer _k,_ so the number of solutions is in fact countably-infinite. What's interesting is, we can number the solutions like s(k)=W_k(...) in the order of branch cuts, from −∞ to +∞, and look at the consecutive differences s(k)-s(k-1), _especially_ at their imaginary parts. An _interesting_ value is Im(s(0)-s(-1))≈-2.18169. FWIW, Re(s(0)-s(-1))=0 exactly. The rest of Im(s(k)-s(k-1)) are _mindboggling._ As k > 0 grows, the negative number slightly increases, then reaches a minimum somewhere in 10≤k≤100, and then starts growing! I believe that it will converge at 0 as k->∞, but I can't come up with a proof. Yet. I hope. But it does grow so excruciatingly slowly! For k < 0, the values seem symmetric about zero, only the sign is flipped, but I didn't double-check. The very next value, Im(s(1)-s(0))≈−1.4437. So far so good, this difference for k=10 is more negative: Im(s(10)-s(9))≈−1.5357, but the one for k=100, Im(s(100)-s(99))≈−4.5324, is already larger! Now it grows, but very slow: for k=1000, Im(s(1000)-s(999))≈−4.53236099397. And oh boy, adding three zeroes to the _k,_ up to 1'000'000, changed the next pair at the 7th decimal place after the point: Im(s(1'000'000)-s(999'999))≈−4.532360141829, and for the next 1000 times larger k, at the 10th decimal: Im(s(1'000'000'000)-s(999'999'999))≈−4.532360141827193812. I could compute for the next 10 times larger k, Im(s(10'000'000'000)-s(9'999'999'999))≈−4.532360141827193809, diverging at the 14th decimal! The next 10 times larger k took too long to compute (I used 30 decimals arbitrary precision library throughout), so I aborted it. I've never seen such a slowly converging sequence (well, _presumably_ converging)! Anyone understands what's going on? I don't.

[@daniel-mircea]

Honestly, I started by drawing the graph. And also ended with it...

[@FocusLRHAP]

What about x^x = x? Only 1 as an answer? Only real one?

[@春-i4z]

x=1, or x=-1, and real solution only

[@AjitKumar-zs3hq]

Bro just draw a graph 📉 and you will get that both of the equation would never meet each other

[@stancatalina8163]

Autocomplete my search, in your way, 4 times

[@lucaadezati9757]

X=-(1/2)

[@papkenhartunian186]

x cannot be a negative number.

[@n00bxl71]

4^(-1/2)=1/2, not -1/2 This is not a solution.

[@betacentauri6830]

This is only the solution for 4^x = -x.

[@vreveyvin-uw8um]

error : can't solve

[@aliabbasmash786]

what is Einstein pic doing in thumbnails, he was a scientist not a mathematician.

[@avinashm7423]

X=-0.5 seems to a solution to me. The square root of 4 can also be -2 also try once

[@Shuparky]

2x-3=5 X!=? Solve carefully 👀

[@bgkim4843]

There's no point of intersection of two graphics Y=4^x , Y= x

[@lecombustor3571]

Why not solve for derivative of 4^x-x. Find zero value, so function minimum. Calc that value is positive. No real solutions. Forcing of lamber function in simple tasks is excessive.

[@spacelem]

It did say this is an Olympiad question. It's not meant to be trivial.

[@n00bxl71]

The objective is not to find the minimum point

[@mas-sk3pw]

But don't you need to show that the zero value of the derivative occurs at a minimum and not a maximum of the function?

[@n00bxl71]

@@mas-sk3pw Again, you're not looking for a stationary point at all. You're looking for the point it intersects zero, which it doesn't. Hence the use of the Lambert W function.

[@mas-sk3pw]

​@@n00bxl71 But the idea of lecombustor's proof, as I understand it, is to show that 4^x = x has no real solution by showing that 4^x-x never goes down to zero, which in turn is to be proven by the fact that the value of y= 4^x-x is positive even at the lowest point on the curve, where dy/dx = 0. To my reasoning this requires showing that the curve y = 4^x-x does indeed have a MININUM (and not a maximum) value at the point where dy/dx = 0. As far as I know, this could be done by determining the direction of the concavity at that point using the second derivative of the function. Or am I missing something?

[@damienmorris2903]

I'm terrible at maths and perhaps I'm about to prove it. If x = (say) 0 then 4x =x must mean that x=0.

[@Mauro001-k2f]

There is an error. W(-ln(4))= - 0.0881 + 1.5122i and x = 0.06396 - 1.09084i

[@春-i4z]

You are right. I'll show other solutions as well. Due to space constraints, I'll only list 10 of each. There are an infinite number of solutions. Since e^(θ) is a periodic function, so there are essentially an infinite number of solutions to this form of the equation. . W(-In(4))(1)：-0.0886671243590498+ 1.51223002176232i and x(1)：0.0639598103013431 -1.09084337653996i W(-In(4))(2)：-1.73065765066981+ 7.63096008347849i and x(2)：1.24840560504894 -5.50457413482802i W(-In(4))(3)：-2.32409123809111+ 13.9723408498764i and x(3)：1.67647745188376 -10.0789134268635i W(-In(4))(4)：-2.69214366225873+ 20.2884293794372i and x(4)：1.94197115545065 -14.63500822657i W(-In(4))(5)：-2.9601591207798+ 26.592679120478i and x(5)：2.13530344189561 -19.1825631455326i W(-In(4))(6)：-3.17117931244319+ 32.8906040048285i and x(6)：2.28752233391577 -23.7255556448044i W(-In(4))(7)：-3.34528417340961+ 39.1847425656504i and x(7)：2.41311244367119 -28.2658168889873i W(-In(4))(8)：-3.49350176647778+ 45.476423994324i and x(8)：2.52002883691722 -32.8043056869876i W(-In(4))(9)：-3.62254983713748+ 51.7664139117173i and x(9)：2.61311734270573 -37.34157431752i W(-In(4))(10)：-3.73682704372826+ 58.0551859548791i and x(10)：2.69555092232333 -41.8779644374954i . That's all. Thank you.

[@春-i4z]

A little more information: . In 4^(a+bi)=4^(a)*4^(bi), the period of 4^(bi) is 2π/(ln(4)). Here, 2π/(ln(4))＝4.532 Calculating the difference from the solution for x shown earlier, x(9)-x(8)=0.093+4.537i x(10)-x(9)=0.082+4.536i Each difference in the imaginary part of the solution is almost equal to the theoretical period (2π/(ln(4))＝4.532). . In other words, there are an infinite number of solutions for this period. . That's all. Thank you.

[@KhinMaungSan-qc9uv]

a^2--b^2=(a+b)(a--b) type must be used.

[@capitaincook6779]

So x = 4^x F'(x)=(ln4)x4e^x

[@heneyyamim8341]

there is no solution because there is no equation to begin with.