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[2024/05/05] Fake power tower (lemmy.world)

submitted 6 months ago by [email protected] to c/[email protected]

6 comments fedilink hide all child comments

Solve x for x^x*x^x^ = 2
Note that the Lambert W function W(x) is the inverse of f(x) = xe^x^

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[–] [email protected] 1 points 6 months ago* (last edited 6 months ago) (1 children)

The text of this post appears wrong on old.lemmy.world. It says "Solve x for x^x*x^x^ = 2" with no superscripts. It appears correctly on lemmy.world.

I assume we're meant to find an expression of W() and square roots and stuff, which expresses an exact answer. Since finding a decimal approximation somewhere between 1 and 2 using a binary search would be too easy.

[–] [email protected] 2 points 6 months ago (1 children)

I believe it is:

spoiler

e^W(W(ln(2))

spoiler

x=W(x)*e^(W(x))

x^(x*x^x)=2
x*x^x*ln(x)=ln(2)
x*e^(ln(x)*x)*ln(x)=ln(2)
u=x*ln(x)
u*e^u=ln(2)
u=W(ln(2))
x*ln(x)=W(ln(2))
e^(ln(x)*x)=e^W(ln(2))
x^x=e^W(ln(2))
x = square-super-root(e^W(ln(2)))
wikipedia says this is equivalent to:
x=e^W(ln(e^W(ln(2))))
but I don't know how they arrive at that.
x=e^W(W(ln(2))

working backwards to verify:
x=e^W(W(ln(2))
ln(x)=W(W(ln(2))
ln(x)*x=W(ln(2))
ln(x)*x*e^(ln(x)*x)=ln(2)
ln(x)*x*x^x=ln(2)
e^(ln(x)*x*x^x)=2
x^(x*x^x)=2

[–] [email protected] 2 points 6 months ago

x^x=e^W(ln(2)) isn't wrong, but it's in a form that's inconvenient to say the least.

Picking up from x*ln(x)=W(ln(2))

spoiler

x^x is a far superior substitution, but it takes a bit to notice it