siriusmart@lemmy.worldM to Daily Maths Challenges@lemmy.world · 5 months ago[2024/05/05] Fake power towerlemmy.worldimagemessage-square6fedilinkarrow-up116arrow-down10file-text
arrow-up116arrow-down1image[2024/05/05] Fake power towerlemmy.worldsiriusmart@lemmy.worldM to Daily Maths Challenges@lemmy.world · 5 months agomessage-square6fedilinkfile-text
minus-squarezkfcfbzr@lemmy.worldlinkfedilinkEnglisharrow-up4·edit-25 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)
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