Do I understand the concept of $x^{0.84}$ correctly?

Question

I'm trying to understand the concept of $x^{0.84}$ that Jeffrey Lagarias found for Collatz Conjecture. If I'm wrong, please correct me with an answer. I understand such that,

Suppose the interval $\left[1, 2^{1\,000\,000} \right]$ is given. In this interval we have at least $\lfloor{(2^{1\,000\,000}})^{0.84}\rfloor=2^{840\,000}$ natural numbers which is goes to $1$. In other words, we can choose at least such $2^{840\,000}$ natural numbers from this sequence $\left\{1,2,3,4,5,6,7,8,9\cdots 2^{1\,000\,000} \right\}$, which goes to $1$. Or, there are at least $2^{840\,000}$ natural numbers in this interval $\left[1, 2^{1\,000\,000} \right]$ (But, we don't know exactly what the numbers are ?), which gives result $1$. So, Collatz Conjecture is correct for at least $2^{840\,000}$ natural numbers. Is my understanding correct?

Answer 1

(OP:)... So, Collatz Conjecture is correct for at least $2^{840\,000}$ natural numbers. Is my understanding correct?

As I've read Lagarias & Krasikov your take is correct. However in your sentence which I've cited here you should have included the clause:

... So, Collatz Conjecture is correct for at least $2^{840\,000}$ natural numbers below $2^{1\,000\,000}$. (...)

Otherwise you come in conflict with the property, that even infinitely many numbers are known to converge to $1$, for instance from the infinite sequence $\{1,2,4,8,\cdots,2^k, \cdots\}$ or from the infinite sequence $\{1,5,21,85, \cdots , {4^k-1 \over 3} , \cdots \}$ or from infinitely many infinite sequences generated in this style.

Answer 2

$x^{0.84} = x^\frac{21}{25} = (x^{21})^\frac{1}{25}$, which is the unique solution $y$ of $y^{25} = x^{21}$. For irrational $y$, $x^y = \sup_{z\in\mathbb{Q}\cap(-\infty,y)}x^z = \inf_{z\in\mathbb{Q}\cap(y,\infty)} x^z$.