By that logic then the logarithm is also cheating. The point is we know how to calculate and get approximate values but we don’t have all day to keep approximating better and better estimates. For example 2^x=3 gives a number greater than 1 but less than 2 for x and we can keep getting better estimates but telling a computer or calculator log base 2 of 3 is much faster than wasting our time.
I raised both sides of the first equation to 1/x to get x on one side. Then with a little algebra came to lnx = -W(ln(sqrt2)). As usual, your approach was more elegant, but it is reassuring to show that the two methods converge to the same result.
This was a very nice equation to solve. However I think also an interesting result is that this also gives us a way to compute W(x), when -1/e ≤ x ≤ e (not sure about convergence, mostly educated guesses here) with W(x) = x * (e^-x)^(e^-x)^(e^-x)... I don't think it's awfully useful, but hey at least there's a new way.
The problem with using Lambert's W function is that it's a magical solution that you don't solve analytically, but with Wolfram Alpha or something like that. As long as I have the magic box, I can get the solution. But if I have access to Wolfram Alpha, I can probably plug in the original equation and get the answer anyway. In short, Lambert's W function is a cool thing, but it doesn't seem to help me understand the math any better. It's just an algorithm in my bag of tricks that yields an answer without giving me any gut-level insight into what I'm doing.
By your logic the logarithm is also magic. You can’t always get a nice answer. For the equation 2^x=3, x is some number greater than 1 and less than 2, which is transcendental. The point is that these function have nice properties and we know methods to approximate these functions very well (Taylor Series, Numerical Integration, Padé Approximations, Newton’s method, etc), and we can tell a computer to approximate these functions instead of wasting our time. In fact, you don’t need more than a calculator to approximate the Lambert W function with sufficient precision.
@@GeoffryGifari I wouldn't say impossible. It might be rational (although I would bet against it). I was just pointing out that the video made no claim that 0.766665 was an exact value.
Definitely it isn't. The equation x² = 2^x have 2 positive solutions x=2 & x=4, 1 negative solution x=-e^-W(ln2/2). So the value of the tower in this case is the additive inverse of the negative solution above.
Lambert's W function always feels like cheating.
By that logic then the logarithm is also cheating. The point is we know how to calculate and get approximate values but we don’t have all day to keep approximating better and better estimates. For example 2^x=3 gives a number greater than 1 but less than 2 for x and we can keep getting better estimates but telling a computer or calculator log base 2 of 3 is much faster than wasting our time.
good point
Learn alot... thanks...❤❤❤.
I raised both sides of the first equation to 1/x to get x on one side. Then with a little algebra came to lnx = -W(ln(sqrt2)). As usual, your approach was more elegant, but it is reassuring to show that the two methods converge to the same result.
Nice!
Nice job!
This was a very nice equation to solve. However I think also an interesting result is that this also gives us a way to compute W(x), when -1/e ≤ x ≤ e (not sure about convergence, mostly educated guesses here) with W(x) = x * (e^-x)^(e^-x)^(e^-x)... I don't think it's awfully useful, but hey at least there's a new way.
The problem with using Lambert's W function is that it's a magical solution that you don't solve analytically, but with Wolfram Alpha or something like that. As long as I have the magic box, I can get the solution. But if I have access to Wolfram Alpha, I can probably plug in the original equation and get the answer anyway.
In short, Lambert's W function is a cool thing, but it doesn't seem to help me understand the math any better. It's just an algorithm in my bag of tricks that yields an answer without giving me any gut-level insight into what I'm doing.
By your logic the logarithm is also magic. You can’t always get a nice answer. For the equation 2^x=3, x is some number greater than 1 and less than 2, which is transcendental. The point is that these function have nice properties and we know methods to approximate these functions very well (Taylor Series, Numerical Integration, Padé Approximations, Newton’s method, etc), and we can tell a computer to approximate these functions instead of wasting our time. In fact, you don’t need more than a calculator to approximate the Lambert W function with sufficient precision.
Does it make sense to have a mathematical object like that which "extends to both sides"?
..... y^y^y^y ......
so there isn't really a "bottom" term?
could this be used in the study of recursions?
0.76666665?
so that infinite square root power tower is rational?
interesting
Nah
Very doubtful that it's rational. The video makes clear that the computed value is only approximate.
@@TedHopp but is it impossible?
@@GeoffryGifari I wouldn't say impossible. It might be rational (although I would bet against it). I was just pointing out that the video made no claim that 0.766665 was an exact value.
That's a solution to the equation x²=2^x
Definitely it isn't. The equation x² = 2^x have 2 positive solutions x=2 & x=4, 1 negative solution x=-e^-W(ln2/2).
So the value of the tower in this case is the additive inverse of the negative solution above.
@@duytuanlengo5038 it is the same just negative
e^(-W(ln2/2))=0,76666
e^(-W(-ln(sqrt(2)/2))
First to comment?
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