2^x = 4x
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- Опубліковано 9 лип 2024
- This problem has two solutions. The second solution, if done by algebra will require a special function called the Lambert W function. In my solution, I used a power series approximation to estimate the W(t)
This is the video I mentioned
• Lambert W Function
'if the value inside, the argument here, is close to 0' when did you become a physicist? 😆
Haha. There's a lot of engineering in me.
Something that is seriously overlooked in your videos are your straight lines when dividing the board! I know its just a side note but you have to admit that Mr Prime draws some of the best straight lines! It is extremely satisfying to see 💯
2^x = 4x = 2²x, therefore x = 2^(x-2). Raising the both sides to the (1/(x-2))-th power, we have x^(1/(x-2)) = 2. Note that if we square both sides, we have x^(2/(x-2)) = 4, or x^(2/(x-2)) = 4^(2/(4-2)). By comparison, x = 4 is a solution. The other one is only possible to get using the Lambert function.
blackpenredpen derived a formula for a^x + bx + c = 0 you can use the formula here too!
yep, I ve done that like that
😮😮😮 can you please tell the formula
@@jimmywatson7950 search bprp solution for transcendal equation
@@jimmywatson7950It's the quadratic formula. (-b +/- sqrt(b^2 - 4ac))/2a
BUT. BPRP did not invent this formula.
The quadratic formula was not derived by bprp.
That’s is actually so cool, great video man
Love your work man:)
Excellent, clearly explained video :)
You celebrate Math. Great to watch.
I literally love his videos ❤
love from Dubai!
Just curious. Instead of using the appropriate approach you did, can’t you just use the actual lambert W function? You’ve shown it a few times. That would get you the number you’re looking for regardless of how close to zero the answer is or not, wouldn’t it?
The most ASMR voice ever!!!
Great video man
Congrats from France
I didnt know there was a formula for the w function wow
It is a formula for an approximation of W, not a formula for W.
I think its the Taylor series of the w function, hence why it only works for small x. You need infinite terms for it to be exact
There's a Taylor expansion for any function. You just have to be able to calculate all the derivatives. It looks like in this case you can continue the series and get a better approximation by adding more and more terms of the form (-1)^(n-1) * n^(n-2) * t^n / (n-1)!. It should be pretty easy to prove since W(x) has a nice expression for its derivative: W'(x) = W(x) / x*(1+W(x)). You can calculate the Taylor expansion around any value too. Not just zero.
@@BangkokBubonagliathere is not a Taylor expansion for any function, though you’re right that there is one for the Lambert W function
If not applying inspection for solution x=4, is it possible to find 4 by algebra via product log function?
You’ll notice at one point he refers to his formula as “the principle branch of the Lambert function.” Just as sqrt(x) gives us only one of the up to two possible solutions for x=φ^2 (thus we sometimes call it “the principle root”) W(x) only gives us one of the possible solutions for x=φe^φ. It is possible to evaluate one of the other solutions (which in this case would be 4), but it would not use this formula which gives us the “principle branch”
Is there any way to find those other lambert w function branches without using product log calculators?
I doubt it
Really awesome, many thanks, Sir!
Why did it blur the phi
Yeah really weird
Yeah really weird
Yeah really weird
If there are multiple solutions, then which solution is achieved by using the Lambert W function? More specifically... In this case can the solution x=4 be achieved using the Lambert W function?
There are multiple branches of Lambert W function. Each branch of Lambert W is represented using W subscript number. And W_0 and W_-1 provide the real solutions
@@CarlBach-ol9zbpiggybacking off of this, you’ll notice that he describes the approximation as giving “the principle branch” which will be the one that any calculator will give you if unspecified. You may or may not have heard sqrt called “the principle root” before. This is because the equation x^2=φ may have more than one solution, but the principle root just gives us the positive solution. In this case you may think of “principle” as meaning the “default” answer, even if it’s not the only one
Pretty sure your voice got muted or something when you were talking about phi sir. Maybe an issue with youtube?
Damn UA-cam policy!! Now I’ll never know what the flower is called!
Phi
@@PrimeNewtonsI figured it couldn’t have been phi for UA-cam to flag it, lol. I can’t imagine what it thought you were saying. Anyways Great video, thank you much!
better way - use iterations ; just start with x = 2^x/4 and put x = 0 , then keep on putting the result values again in the expession till the value of x is almost equals to the expresison of 2^x/4 ; that'd be your answer
Well, that would be numerical rather than analytical.
@@TheFrewah since u already know there are 2 solutions , one is 4 and other is somewhere near 0 , better to solve like this instead of going to wolframalpha for W values
@@the_real_nayak In practice it may be if you havethis problem as an engineer
Thank you sir. I have two questions. (1) How can we determine the number of real roots ? (2) Can we get the solution x=4 from Lambert W function method ?
The Lambert function calculator gives two solutions for (-ln 2) / 4 :
- 2.772589 and - 0.214811
These two solutions divided by - ln 2 will give us 4 and 0.3099... .
(1): 2^x-4x is positive for x< 0, negative for x=3 and positive for x=5. ==> there must be at least two real roots. The second derivative of 2^x-4x is positive for all values of x ==> there are at most 2 real roots.
(2) Yes: x=4 is the solution on the second branch of W. (c.f. Wikipedia article on Lambert W function, and Prime Newtons excellent clip, link in the description of this video).
You can use calculus to figure that out. Set f(x)=2^x-4x. Then the first derivative is f'(x)=ln2*2^x-4. To assess for maximum or minimum points we set the first derivative to 0. ln2*2^x-4=0. This equation is easily solvable, 2^x=4/ln2,
x=ln(4/ln2)/ln2. The second derivative is f''(x)=(ln2)^2*2^x. This is positive for all real x, therefor x=ln(4/ln2)/ln2 is a local minimum. The value for f for this minimum is negative. However for x=0 f(x) is positive and for x=5 f(x) is positive as weOur minimum is in between these two values and this is the only extreme point we have as f'(x) can become zero only at this one point. Therefor our equation must have exactly 2 solutions.
@@SG49478 The reasoning is mostly correct, but it is not sufficient proof that we have only two solutions. By trial, two values were found for which the function is positive, 0 and 5, but this is not part of the demonstrative mathematical rigor that is required. My view is that one cannot show that there are only two solutions except by actually solving the ecuation f(x) = 0.
@@marianl8718 Well then explain to me how a steady function with exactly one local minimum where f(x) is negative at that minimum and no local maximum and two values where one is smaller and one is greater than the x value of the local minimum with each of them with f(x) being positive could have by any means more than 2 zero points. That is simply not possible.
If the graph turns around and cuts the x-axis a third time, the function would have to have at least one local maximum. However with the first and the second derivative we have proven, that this function can not have a local maximum. Therefor in my opinion the proof is sufficient.
9:19 "Come on!"
Is there any way to get a value for x in the equation 3^x^x = 10? Just like to know because the only way I gotten a value was from a graphing calculator.
3^x^x=10
Let x^x=y
x.ln x=ln y
ln x.e^ln x=ln y
W(ln x.e^ln x)=W(ln y)
ln x=W(ln y)
x=e^W(ln y)
Now 3^x^x=3^y=10
y.ln3=ln10
y=ln10/ln3
x=e^W(ln(ln10/ln3))
x~1.5918
@@kemosabe761 thanks!
Of course, you could just use the solver function on your calculator, but where's the fun in that 🎉
why did you round to 0.309? The actual answer according to wolfram alpha is 0.3099 which rounds to 0.310
I didn't hear round (could have missed it) but he could have truncated it to estimate. Also when he wrote it on the board it was from an estimated method and he said the exact answer from the calculator was 0.31
@@vecenwilliams8172 ahh ok
x≈0,30990693238069
Why does the W function not give x=4 as the solution?
Actually, it does give x=4. W is a multivalued function. For x between -1/e and 0 there are two real branches W_0 and W_-1. In the clip, he shows the solution for the first branch. x=4 is the solution for the second branch. For details read the article on "Lambert W function" in Wikipedia. Also I recommend Prime Newton's clip about the W function (link is in the description of this clip).
How quickly would we have gotten to a reasonable answer just using Newton’s method from the start?
That wouldn’t be a mathematical way, it would be a numerical method. This channel os about math
@@TheFrewah True, but he just used a truncated power series to estimate a numerical solution to the product-log function.
@@sciphyskyguy4337 Yes but still analytical, power series is what you end up with if you want to calculate e to a high degree of decimals.
Newton-Raphson is based on a Taylor series expansion and has a region of convergence. Sounds pretty analytic to me. :-)
Gostei muito e obrigado
just divide both sides by zero
How’s the approximation of the function found looks like some Taylor series stuff
X=W(Ln(4th root of 2))/-Ln(2)
Don't have access to the internet, but can watch on youtube 🎉🎉
Did it by contoured method and solutions coming are 4 and approximately 0.309905
2^×=4^×
>>2^×-4^×=0
2^×(1-2^×)=0
1=2^×
X=0
X⁰=1
X=4 is the answer
Take log base 2 in both side and solve further
Prove the MacLaurin expansion of the lambert function next
"1? 2? 3? 4? Ye 4."
Omg… Is it a BLACKMATH???
2^4=4*4 x=4 I didn’t graph use a calculator or anything I did it in my head.
❤
four
5:26 "let's not call it x, let's call it... x"
I had to go back and make sure I heard him right lol
4
W function
x=4
By inspection, x=4 as 2^x=2⁴=16 and 4x=4(4)=16
There is a math error at 9:10. You forgot to divide the LHS by 4. So the solution is not -W(-ln(2))/ln(2) but -4 * W(-ln(2)) / ln(2)
This use of a case specific function to get a numerical approximation seems to support my suspicion that maths is a branch of engineering ;)
X =4. I did it in my mind.😅
4