Intervals with Transcendental Outputs

[Hertz]

Maybe I'm getting a bit ahead of myself here, but for the sake of curiosity I'll ask it anyways. Is there any way that you can test when a function h(x) will have transcendental outputs vs rational or algebraic outputs?

Specifically, if I have the following function
f(x) = ssrt(x)

Where ssrt is the inverse of a second degree power tower (ssrt(x^x) = x).

Meaning, for a given value of x, the ssrt() function will give me the number that when raised to the power of itself equals x.

Good examples of this function are ssrt(4) = 2 because 2^2 = 4 and ssrt(27) = 3 because 3^3 = 27.​

Would it be possible to find the x intervals that make f(x) transcendental?

 

[fresh_42]

I doubt that there are intervals as solution. Generally you should look up Lambert's W-function and the Gelfond-Schneider theorem.