Question Number 552 by 123456 last updated on 25/Jan/15
$${proof}\:{that}\:\mathrm{log}_{\mathrm{2}} \mathrm{3}\:{is}\:{irrational} \\ $$
Answered by prakash jain last updated on 25/Jan/15
$$\mathrm{Let}\:\mathrm{us}\:\mathrm{say}\:\mathrm{log}_{\mathrm{2}} \mathrm{3}\:\mathrm{is}\:\mathrm{rational},\:\mathrm{then} \\ $$$$\mathrm{log}_{\mathrm{2}} \mathrm{3}=\frac{{p}}{{q}},\:{p},{q}\in\mathbb{N} \\ $$$$\mathrm{3}=\mathrm{2}^{\frac{{p}}{{q}}} \Rightarrow\mathrm{3}^{{q}} =\mathrm{2}^{{p}} \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{not}\:\mathrm{possible}.\:\mathrm{Hencr}\:\mathrm{log}_{\mathrm{2}} \mathrm{3}\:\mathrm{is}\:\mathrm{irrational}. \\ $$