- Solve x for x^x*x^x^ = 2
- Note that the Lambert W function W(x) is the inverse of f(x) = xe^x^
this post was submitted on 06 May 2024
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solution
x^(x*x^x) = 2(x^x)^(x^x) = 2
k = x^x
k^k = 2
k*ln(k) = ln(2) → Log of both sides
ln(k) * e^ln(k) = ln(2) → k = e^ln(k)
f(ln(k)) = ln(2)
ln(k) = W(ln(2))
ln(x^x) = W(ln(2))
ln(x)*e^ln(x) = W(ln(2)) → Same step as noted earlier
f(ln(x)) = W(ln(2))
ln(x) = W(W(ln(2))
x = e^W(W(ln(2)))
x ≈ 1.3799703966 (via Wolfram|Alpha, seems to be the correct value)
ggs