prove that ln(x) is irrational for x natural


Question and Answers Forum

All Questions Topic List
None Questions
Previous in All Question Next in All Question
Previous in None Next in None
Question Number 40434 by vitlu last updated on 21/Jul/18

$p r o v e t h a t \ln (x) i s i r r a t i o n a l f o r x n a t u r a l$
Commented by math khazana by abdo last updated on 21/Jul/18

$l e t x = n n a t u r a l > 1 l e t p r o v e t h a t l n (n) \notin Q$ $i f l n (n) \in Q \exists (p, q) \in N^{2} / l n (n) = \frac{p}{q}$ $w e c a n t a k e D (p, q) = 1 \Rightarrow n = e^{\frac{p}{q}} \Rightarrow n^{q} = e^{p} b u t$ $e^{p} = \sum_{k = 0}^{\infty} \frac{e^{k p}}{k!} \notin N s o t h e e q u a l i t y n^{q} = e^{p} i s$ $i m p o s s i b l e b e c a u s e n^{q} \in N a n d e^{p} \notin N .$
Commented by vitlu last updated on 21/Jul/18

$t h a n k s$
Commented by maxmathsup by imad last updated on 21/Jul/18

$y o u a r e w e c o m e$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum