Question

Comparati nr x si y daca au raportul 14^46
_____
7^69 repede

Answer 1

$14^{46} \,\,\square \,\,7^{69}\\ (2\cdot 7)^{46}\,\,\square \,\,7^{69}\\ 2^{46}\cdot 7^{46}\,\,\square \,\,7^{69}\Big|:7^{49} \\2^{46} \,\,\square \,\,7^{20}\Big|\sqrt{}\\ 2^{23} \,\,\square \,\,7^{10}\\ \log_{2}2^{23} \,\,\square \,\,\log_{2}7^{10}\\23 \,\,\square \,\,10\log_{2}7\\2,3 \,\,\square \,\,\log_{2}7\\\\ \log_{2}7>2,5 \Rightarrow 2\log_{2}7 > 5\Rightarrow \log_{2}7^2 >\log_{2}2^5\Rightarrow\\ \\ \Rightarrow \log_{2}49 >\log_{2}32\,\,(A)\Rightarrow \log_{2}7 >2,5>2,3\\ \\ \Rightarrow 2,3 \,\boxed{<} \,\log_{2}7 \Rightarrow \boxed{14^{46} <7^{69}}$