• Solve x for xx*x^x = 2
  • Note that the Lambert W function W(x) is the inverse of f(x) = xex
  • zkfcfbzrEnglish
    4•6 months ago
    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)