Question

log^2 (2) (cos^2x)-22log2(cosx)-12=0 решить

Answer 1

$log^2_2 (cos^2x)-22cdot log_2(cosx)-12=0$

ОДЗ : cos x > 0

$log^2_2 (cos^2x)-11 cdot log_2(cos^2x)-12=0$

Замена переменной $y = log_2(cos^2x)$

$y^2 - 11y -12 = 0~~~Leftrightarrow~~~ (y-12)(y+1)=0\ \ 1)~~y-12 = 0;~y=12\ ~~~log_2(cos^2x)=12; ~~cos^2x=2^{12}=4096\ ~~~cos^2xleq 1;~~~Rightarrow~~~x in varnothing\ \ 2) y+1 =0;~~y=-1\ ~~~log_2(cos^2x)=-1; ~~cos^2x=2^{-1}=dfrac{1}{2}\ \ ~~~cos~x=pm dfrac {1}{sqrt{2} };~~cos~x>0~(ODZ) ~~Rightarrow\ \ ~~~cos~x= dfrac {1}{sqrt{2} };~~boxed{boldsymbol {x=pm dfrac{pi}{4}+2pi n; ~nin Z}}$