Question

√(log⅓log₄(|x|²-5))
Domain?

Answer 1

function y = √{log⅓log4(|x|² - 5) }

for log to be defined , when,
( |x|² - 5) > 0
log4( |x|² - 5) > 0
and according to square root ,
log⅓log4(|x|² - 5) ≥ 0

now, solve all individually

( |x |² - 5) > 0
(|x | - √5)(|x | + √5) > 0
|x | > √5 or | x | < -√5
but we know ,
|x | is always positive so, |x | < -√5 isn't possible.
hence, |x | > √5 => x> √5 or x< -√5


log4(|x|² -5) >0
|x|² - 5 > 1
|x|² - 6 > 0
similarly here, x> √6 and x < -√6

log⅓ log4( |x|² - 5) ≥ 0

base of log is fraction and less then 1 so, after removing log inequality sign change .

log4( |x|² - 5) ≤ 1

(|x |² - 5) ≤ 4

(|x|² - 9) ≤ 0

(|x| - 3)(|x| + 3) ≤ 0

-3 ≤ | x | ≤ 3
but |x| is positive so,
0 < |x| ≤ 3
here two cases , possible

case1:- |x| > 0 this true for all x €R - {0 }

case2 :- | x | ≤ 3 => -3≤ x ≤ 3

hence, -3 ≤ x < 0 and 0 < x ≤ 3

now take intersection of all coming value of x of condition by using number line

hence, x€( -3, -√6) U ( √6, 3)