Simple Yet Difficult. How to Solve Using Lambert W Function

[@XanderAnimations]

Fun fact: e^W(lnx) is the formula to find the *Super Square Root!* Notice how x^x is a tetration equation, it's x tetrated to 2 or x^^2. The solution to x^^2 = a is the super square root (ssqrt) of a, or e^W(lna)

[@jim2376]

Cool beans! Never knew about the Lambert W function. This was a very helpful introductory video to the function. Thanks.

[@homoexhumo]

I set x=1 and n=2 Then I apply the recursive relationship: x=(n^(1/x) + x)/2 As it stands, this method converges fairly quickly to the more general problem x^x = n for some range of real values ​​of n. It has been many years since I worked on this problem, and I don't remember much of the details, except that the above method undergoes many changes depending on the range of values ​​of n. The boundaries of these values, if I remember correctly, depend on the constant e. In general, this method and its variants also work for complex values ​​of n.

[@ayushoriginal]

Simply assume x = 2 Thus square root both side you get the answer... Which is x = √2 I.e 1.414 approximately

[@seroujghazarian6343]

x^x=sqrt(2)^sqrt(2) can't be 2 as by using the Gelfond-Schneider theorem, we can prove that sqrt(2)^sqrt(2) is transcendental

[@marlydahl]

So, that's what we call magic!

[@buckyjennings8854]

What is the point if you still need a computer program to give you the value?

[@NichaelCramer]

@@buckyjennings8854 : I was concerned about this initially as well, but rather than thinking of it as “needing a computer”, think of it “as needing a table of the values of the W function”. In other words this is no different from, for example, if the solution were of the form “X = ln(5)” or of the form“X = cos(5dg)” (or, for that matter, “x = sqrt(5)”). In all these these cases you’ve just expressed your answer in terms of a constant value of a previously well-defined function.

[@souzasilva5471]

How do you find 16 in Lambert's formula? Not how you find the indices 0 and -1 of W.

[@Gezraf]

how is that difficult tho