Prove by absurd that log 2 is the number irrational


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Question Number 159171 by LEKOUMA last updated on 13/Nov/21

$P r o v e b y a b s u r d t h a t \log 2 i s t h e$ $n u m b e r i r r a t i o n a l$
	Answered by mr W last updated on 14/Nov/21

	$s a y \log 2 = \frac{p}{q} w i t h p, q \in Z$ $2 = 10^{\frac{p}{q}}$ $2^{q} = 10^{p}$ $t h e l a s t d i g i t o f 2^{q} c a n o n l y b e 2, 4, 6, 8,$ $b u t t h e l a s t d i g i t o f 10^{p} i s a l w a y s 0 .$ $\Rightarrow c o n t r a d i c t i o n!$ $o r$ $2^{q} = 2^{p} 5^{p}$ $2^{q - p} = 5^{p}$ $e v e n = o d d$ $\Rightarrow c o n t r a d i c t i o n!$
	Commented by gsk2684 last updated on 14/Nov/21

	$n i c e$
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