The text of this post appears wrong on old.lemmy.world. It says “Solve x forx^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.
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
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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.
I believe it is:
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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)=2x^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
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x^xis a far superior substitution, but it takes a bit to notice it