What is a NaturaW Word™?

This is in the spirit of the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles.

If a word conforms to a special rule, I call it a NaturaW Word™.
Use the following examples below to find the rule.

$$% set Title text. (spaces around the text ARE important; do not remove.) % increase Pad value only if your entries are longer than the title bar. % \def\Pad{\P{0.0}} \def\Title{\textbf{ NaturaW }} % \def\S#1#2{\Space{#1}{20px}{#2px}}\def\P#1{\V{#1em}}\def\V#1{\S{#1}{9}} \def\T{\Title\textbf{Words}^{\;\!™}\Pad}\def\NT{\Pad\textbf{Not}\T\ }\displaystyle \smash{\lower{29px}\bbox[yellow]{\phantom{\rlap{rubio.2019.05.15}\S{6px}{0} \begin{array}{cc}\Pad\T&\NT\\\end{array}}}}\atop\def\V#1{\S{#1}{5}} \begin{array}{|c|c|}\hline\Pad\T&\NT\\\hline % \text{ ACID }&\text{ SWEET }\\ \hline \text{ HIGH }&\text{ LOW }\\ \hline \text{ DICE }&\text{ ROLL }\\ \hline \text{ CRY }&\text{ LAUGH }\\ \hline \text{ DOES }&\text{ DONE }\\ \hline \text{ EDGE }&\text{ SIDE }\\ \hline \text{ FAX }&\text{ PHONE }\\ \hline \text{ BIRDED }&\text{ BIRDING }\\ \hline \text{ WEB }&\text{ LINE }\\ \hline \text{ BIG }&\text{ SMALL }\\ \hline \end{array}$$

And, if you want to analyze, here is a CSV version:

NaturaW Words™,Not NaturaW Words™
ACID,SWEET
HIGH,LOW
DICE,ROLL
CRY,LAUGH
DOES,DONE
EDGE,SIDE
FAX,PHONE
BIRDED,BIRDING
WEB,LINE
BIG,SMALL

Hint 1:

W = 10K

Hint 2:

• Is "Birded" a word? – Brandon_J Aug 15 at 14:37
• @Brandon_J p.p. of the verb "bird" :P thefreedictionary.com/bird – Conifers Aug 15 at 14:41
• Ah. Very nice. I was thinking of birding as a noun and got thrown off. – Brandon_J Aug 15 at 14:42
• I can't stop seeing rot13(oveqfrrq, synk, naq frqtr) in there. – hdsdv Aug 15 at 19:53

Considering hints:

Seems like words written as numbers (A=1, B=2, etc.) appears in first 10000 digits of Euler's number (Hint 1).
I checked it here
So words as numbers that are there: 1394, 8978, 4935, 31825, 415519, 5475, 6124, 2918454, 2352, 297 (BIRDED is in link as "291845 4", notice space)
And that are not there: 19235520, 121523, 18151212, 1212178, 415145, 19945, 16815145, 291849147, 129145, 191311212
Hint "log" and Name "natura" means Natural logarithm which has base of Euler's number

• You've found it~ :P – Conifers Aug 22 at 12:14