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 : )

Is there a way to approximate that easily?