I am currently a bit stuck trying to solve this problem only using the Lambert W function and elementary functions.
I succeeded in solving simpler functions like y = x^x and y = x^(x^2), but I cannot seem to solve this one.
Here is my current best attempt (based off of the steps for x^(x^2)):
y = x^(x^x)
ln(y) = x^x * ln(x)
= x^x * ln(x^(x / x))
= x^x * (1/x * ln(x^x))
= 1/x * x^x * ln(x^x)
x * ln(y) = x^x * ln(x^x)
= ln(x^x) * x^x
= ln(x^x) * e^(ln(x^x))
= x*ln(x) * e^(x*ln(x))
x*ln(x) = W(x * ln(y))
x*e^x = e^W(x * ln(y))
x = W(e^W(x * ln(y)))
But this isn't particularly useful as x is still in the formula.
I could solve it if I knew how to solve ln(y) = 1/x * x^x * ln(x^x).
Can you please help me?
I tried it myself a while ago for solving x^x^x=2 but I had no luck. However, we can use the super cube root : )