What do you think about this problem? If you're reading this ❤️. Hello My Friend ! Welcome to my channel. I really appreciate it! @higher_mathematics #maths #math
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.
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.
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.
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.
@@shashwatraj2381 As you progress in math, you will find it necessary to read and compute complex material. Check out the works of Srinivasan Ramanujan.
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.
@@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.
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.
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."
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.
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.
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
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 ;-)
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).
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.
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 .
@@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.
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.
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
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.
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.
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.
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.
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
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
Hello, Why Sightsout all Questions like int Mainschoolniveau Perhaps, int Realschool ist Mathsquestion Ten Times Longer Perhaps. Myselfs didn’t Understand Why ITS‘S. Lg Sven BALZER
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.
😂😂😂😂 will solving this maths in school provide a mansion and wealth like Jeff bezos 😂😂. Waist of time and brain.. in my country Nigeria we are only intelligent in making Dollars and working harder.. mostly eastern parts of Nigeria 😅
We were promised a beautiful equation. What we got was a monstrosity with no apparent analytical solution, instead requiring some obscure function to be defined. Are math Olympiad people supposed to have written the Lambert W function in their solution? Or somehow calculate the approximate solution with numerical methods? Venting after spending 4 hours on this and finding there was no solution that could be done by hand after all 🤡
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.
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.
@@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.
crap,the curve y = 4 exp. x has no point in common with the line y = x therefore the task is unsolvable.Never don't solve tasks that are unsolved /becauce you want to be clever/.the end
Una reverenda estupidez esa "ecuación " no tiene solucion y esto se puede demostras de manera simple .. solo hay dos posibles exponentes que puedan satisfacer esa ecuacion y es 1 y 0 .. 4^1= 4 Y 4^ 0 =1 ESTO LO PUEDE VERIFICAR O RESOLVER UN ESTUDIANTE DE PRIMARIA ...
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
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.
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.
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?
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"!
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.
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
No, the guy doesn't even understand it himself, which is why he said a lot of rubbish. I tried to follow his calculations, but I was lost too. Even though I already knew about this product log function.
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.
@@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.
@@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?
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.
Beside the math did anyone try to remove hair from screen like me in beginning?😂😂🤣
It's not a hair. It's a plutonium shaving.
@@RadicalCaveman 😂🤣😂🤣😂🤣🤣
I'm 63. I have a PhD in engineering. I've never heard of Lambert's W function.
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
@@assiya3023عمري 40 عامًا تقريبًا وأنا⚡ مهندس برمجيات. التعلم هو الطريق.
I hadn't either, BS in engineering in '83.
Me too. Electronic Engineer '07
Same here 42 years old,but it's always good to learn
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.
Not only that, but how one finds complex solutions involving the Lambert W function. He pulled that from somewhere.
Hello, It’s Only forn Part Highierer Like int Realschoolniveau Perhaps. Formytaste Is This Not Highschool or Studyniveau.
Lg Sven BALZER
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.
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.
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.
You nailed it mate
Tldr?
@StatisticalLearner I agree with @javgrzg, excellent elaboration on the Lambert W function and Euler’s and Dirac’s work, thank you.
@@shashwatraj2381 As you progress in math, you will find it necessary to read and compute complex material. Check out the works of Srinivasan Ramanujan.
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.
...and so 4ᵗʰ.
How do you find complex roots?
@@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.
Take it easy on him. I don't think he understands what the Omega function is yet.
@@papkenhartunian186 research the Lambert W function, that is the solution.
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.
I'd stick the two functions in Excel and have a look at where, if at all, they cross.
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."
Somehow I do not feel enlightened
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.
No real solution but it has complex solution as explained in the video.
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.
je confirme !
How did you find your answer?
Bravo!
4^(1.248-5.505i) = 1.24446 - 5.50222 i
@@papkenhartunian186 Newton's method.
z(n+1) = z(n) - f(z(n))/f'(z(n))
You have explained an answer that does not exist. You have not explained how you found the complex solution.
You speak too fast!
LOL
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
This is painful, the most trivial inequalities will show that there is no real solution. Higher mathematics?
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 ;-)
this is not higher mat .
ua-cam.com/video/bHKbhgBABXw/v-deo.html
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).
Could you explain how you found the complex root?
G**gle Scipy lambert. There's a good explanation and example how to find complex roots
You keep repeating "Natural log Natural log..."). It is probably your verbose but some might find it confusing. Dr. Ajit Thakur (USA).
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.
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 .
This question itself wrong. How it is possible to X is more bigger than X 😂😂😂.
X=4^x impossible.
Exactly.
when mathematics becomes impractical,
it essentially becomes a useless waste of time .
@@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
Not if x is 0
@@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.
Lambert W function is confusing
Is it of any practical use or serves as brain teaser only?
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.
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
so u input x=1/2 and output x=2? lol this isn't c++
error : can't solve
The numeric solution he gives is wrong
Every time i can't get into sleep then i choose ur video😮
Wait, I think I figured it out… X = X lmao 😂
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.
The graphs of the functions y=x and y=4^x do not intersect.
Which is why the solutions are complex.
@@jstarks123 R2 is isomorphic to the complex plane ergo the graphs do not intersect there either. The only ‘solution’ is bogus.
@@padraiggluck2980y=x²+1 and y=0 do not intersect in R² either, and yet we know that intersects at the points (±i,0).
@@spacelem x^2+1 has nothing to do with the given problem.
@@padraiggluck2980 except that it fits the same properties you gave for saying "the only 'solution' is bogus".
2^x = x , if we put x = 0 + 1i then 0 + 1i ≈ 1 ,
What a boring explanation....
Needing only one line !!!!
a^2--b^2=(a+b)(a--b) type must be used.
I feel Lambert W is just a cheat😂😂
Sure, but so is the square root function :)
so are the complex numbers lol. They were literally invented to solve the equation x^2 = -1
Bro just draw a graph 📉 and you will get that both of the equation would never meet each other
"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.
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.
There is an error. W(-ln(4))= - 0.0881 + 1.5122i and x = 0.06396 - 1.09084i
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.
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.
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 😂!
كيف يتم حساب. ( ) W
رد ❤
As usual, I would prefer using the graph approach. Since the answers are approximates, a graph works a lot simpler. 😊
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.
Why isn't there a simpler way to solve it using simple algebraic formulas?
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
You take time to explain the most rudimentary algebraic steps, but fly over the use of W. Come on...
Yes he explained middle school math steps but brushed over Lambert W and in particular finding a complex solution.
Such a task will never appear in reality and such an answer is of no use to anyone, because what to do with it?
of two poss. answers i find here.
1 is.
4x.
2 is.
4x =x
x= ?..
Es gibt keine reelle Lösung. Das wird durch eine Kurvendiskussion von f(x) = 4^x - x deutlich.
Can you please explain this statement. I don't understand
Parachuted responses.
What is w fonction, in which field you've applied ln on x...... Etc
What about x^x = x? Only 1 as an answer? Only real one?
x=1, or x=-1, and real solution only
This is NOT a math olympiad question. This is the opposite of a math olympiad question.
ha ha ha ha ha ha
@@leolacic9442 What's funny?
Can't we suppose that x is infinity?
Why ln and not log2 or log4 is used.
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
Hello, Why Sightsout all Questions like int Mainschoolniveau Perhaps, int Realschool ist Mathsquestion Ten Times Longer Perhaps. Myselfs didn’t Understand Why ITS‘S. Lg Sven BALZER
Another stupid question from this channel with the stupid Lambert's W function. Just change your channel name to "Lambert's W function".
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.
Why not using log
I thought about that at first. But after trying to solve it I couldn't find an answer
Autocomplete my search, in your way, 4 times
😂😂😂😂 will solving this maths in school provide a mansion and wealth like Jeff bezos 😂😂. Waist of time and brain.. in my country Nigeria we are only intelligent in making Dollars and working harder.. mostly eastern parts of Nigeria 😅
We were promised a beautiful equation. What we got was a monstrosity with no apparent analytical solution, instead requiring some obscure function to be defined. Are math Olympiad people supposed to have written the Lambert W function in their solution? Or somehow calculate the approximate solution with numerical methods?
Venting after spending 4 hours on this and finding there was no solution that could be done by hand after all 🤡
Isn't it easier to draw graphs
mathematics: no room for error
higher mathematics: *excist*
Was there ever a mathematician with a sense of humor? OK, Einstein.
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...
Yeah. That was weird.
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.
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.
@@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.
@@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.
crap,the curve y = 4 exp. x has no point in common with the line y = x therefore the task is unsolvable.Never don't solve tasks that are unsolved /becauce you want to be clever/.the end
This guy is an absolute fraud. He overcomplicates things to look smart, not realising he's a teacher
Una reverenda estupidez esa "ecuación " no tiene solucion y esto se puede demostras de manera simple .. solo hay dos posibles exponentes que puedan satisfacer esa ecuacion y es 1 y 0 .. 4^1= 4 Y 4^ 0 =1 ESTO LO PUEDE VERIFICAR O RESOLVER UN ESTUDIANTE DE PRIMARIA ...
No solution for real number.
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
Mr. Lambert enters the room.
Stop the over explanation. This should not take 10 mins
Bro why the hate? Its better to give more explanation than not giving any at all
More explanation = More money 🤑
Наши русские хитрозадые математики уложились бы в минуту)
I love this explanation at all .Don't judge wasting descriptions your time because you've already known .
..searching money conten.....
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.
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.
Why Einsten he persanaly admited that he wasnt good im math his first wife was she teached him to pass exams on university
You can also solve this using an infinitary expression and finding the fixed point.
It is obvious that the left side is always greater than th right size. Please stop with these trivial questions!
There's no point of intersection of two graphics
Y=4^x , Y= x
Answer can be infinity value of x 😂
Can you solve this equation: 9x square + 21x -8=0
Interesting. Ln x =1
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?
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)
2x-3=5
X!=?
Solve carefully 👀
what is Einstein pic doing in thumbnails, he was a scientist not a mathematician.
The solution can be easier determined by grafic
Too much explanation of simple stuff. Not enough explanation of hard stuff.
haven't you studied Cambridge syllabus? simply use iteration method. its done
Why do you keep saying natural log natural log four?
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"!
I'M HERE TO TELL YOU THAT IN THE HISTORY OF MATHEMATICS THIS IS AN IMPOSSIBLE EQUATION
ayo imo problems are infinitely more harder than this
why Einstein's picture neer this..was he a stupid 😮
The minimum value of lnx/x is 1/e
I have solved this one, sometime ago very easily. This does not belong to higher mathematics.
Nice reminder. But could you point me to some material explaining the W function?
Wikipedia. Lambert W function.
Thanks, @@LelekKozodoj69
X=-0.5 seems to a solution to me. The square root of 4 can also be -2 also try once
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.
X=2 then? Thats what got without the messing..
Этого можно решать графический легко, и решение нет,зачем столько воды
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
А не проще ли было графики построить?
I don't understand you talk too fast
No, the guy doesn't even understand it himself, which is why he said a lot of rubbish. I tried to follow his calculations, but I was lost too. Even though I already knew about this product log function.
The result I calculated using the AI program is: around X≈0.05
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.
It did say this is an Olympiad question. It's not meant to be trivial.
The objective is not to find the minimum point
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?
@@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.
@@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?
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.
Не видели пациентов из психушку? 4^(-0.5i)=0.5i. не надо мучить голову.