Question

$log _{0,5} ^{2} (-log _{3}x )-log _{0,5} log _{3} ^{2} x \leq 3$

Answer 1

{x>0
{log(3)x<0⇒x<1
x∈(0;1)
log²(0,5)(-log(3)x)-2log(0,5)(-log(3)x)≤3
log(0,5)(-log(3)x)=t
t²-2t-3≤0
t1+t2=2 U t1*t2=-3
t1=-1 U t2=3
-1≤t≤3
{log(0,5)(-log(3)x)≥-1⇒-log(3)x≤2⇒log(3)x≥-2⇒x≥1/9
{log(0,5)(-log(3)x)≤3⇒-log(3)x≥1/8⇒log(3)x≤-1/8⇒x≤$1/ \sqrt[8]{3}$
x∈(0;$1/ \sqrt[8]{3} ]$ U [1/9;1)