Find the intersection point of an infinite power tower and a primorial

[Saracen Rue]

TL;DR

Find the point of intersection between $f(x) = {^{\infty}x} = x \uparrow \uparrow \infty$ and $g(x)=p_{x}$#

Consider $f(x) = {^{\infty}x} = x \uparrow \uparrow \infty$ and $g(x)=p_{x}$#, where $p_x$# is the primorial function and is defined such that $p_n$# is the product of the first $n$ prime numbers. For example, $p_{4}$# $= 2×3×5×7=210$

Let the point of intersection be defined as $(m, f(m))$; determine the value of both $m$ and $f(m)$ correct to five significant figures.

 

[mitochan]

I think power tower is defined and finite for e^{-e}<x<e^{1/e}. How g(x) is defined for x in this region?

 

[TeethWhitener]

How do you define the primorial for non-integers?

 

[Saracen Rue]

How do you define the primorial for non-integers?

Im looking for the best estimation using a curve fitting.