Question

СРОЧНО!!!! 35 БАЛЛОВ!!! ПОЖАЛУЙСТА!!!!
${x}^{ log_{4 }(x - 2) } = {2}^{3( log_{4}(x - 1)) }$
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Answer 1

ОДЗ : x > 0

$x^{\log_{4}x-2 }=2^{3(\log_{4}x-1)}} \\\\\log_{4}x^{\log_{4}x-2 }=\log_{4}(2^{3(\log_{4}x-1)}} )\\\\(\log_{4}x-2)\cdot\log_{4} x=[3(\log_{4}x-1)]\cdot\log_{4}2\\\\\log_{4}^{2}x-2\log_{4} x=(3\log_{4}x-3)\cdot\frac{1}{2}\\\\2\log_{4}^{2}x-4\log_{4} x-3\log_{4}x+3=0\\\\2\log_{4}^{2}x-7\log_{4} x+3=0$

$\log_{4}x=m\\\\2m^{2}-7m+3=0\\\\D=(-7)^{2} -4\cdot2\cdot3=49-24=25=5^{2}\\\\m_{1}=\frac{7-5}{4}=\frac{1}{2} \\\\m_{2}=\frac{7+5}{4}=3\\\\1)\log_{4} x=\frac{1}{2} \\\\x_{1}=4^{\frac{1}{2} } =2\\\\2)\log_{4} x=3\\\\x_{2}=4^{3} =64\\\\Otvet:\boxed{(2 \ ; \ 64)}$