2-log-0-5-2-x-x-log-0-5-x-gt-2-5-

Question Number 4615 by love math last updated on 14/Feb/16

2^{{l o g}_{0.5}^{2} x} + x^{{l o g}_{0.5} x} > 2.5

Answered by Yozzii last updated on 14/Feb/16

R e a r r a n g i n g t h e g i v e n i n e q u a l i t y

l e a d s t o x^{{l o g}_{0.5} x} > 2.5 - 2^{{l o g}_{0.5}^{2} x} .

T a k i n g l o g a r i t m s t o b a s e 0.5 o n b o t h

s i d e s l e a d s t o {l o g}_{0.5} x^{{l o g}_{0.5} x} > {l o g}_{0.5} (2.5 - 2^{{l o g}_{0.5}^{2} x}) .

B y t h e p o w e r r u l e \Rightarrow ({l o g}_{0.5} x) ({l o g}_{0.5} x) > {l o g}_{0.5} (2.5 - 2^{{l o g}_{0.5}^{2} x})

\Rightarrow {l o g}_{0.5}^{2} x > {l o g}_{0.5} (2.5 - 2^{{l o g}_{0.5}^{2} x})

L e t u = {l o g}_{0.5}^{2} x . ∴ u > {l o g}_{0.5} (2.5 - 2^{u})

\Rightarrow 0 . 5^{u} > 0 . 5^{{l o g}_{0.5} (2.5 - 2^{u})}

\frac{1}{2^{u}} > \frac{5}{2} - 2^{u} \Rightarrow 1 > \frac{5}{2} \times 2^{u} - {(2^{u})}^{2}

{(2^{u})}^{2} - \frac{5}{2} (2^{u}) + 1 > 0

2 {(2^{u})}^{2} - 5 (2^{u}) + 2 > 0

(2 (2^{u}) - 1) (2^{u} - 2) > 0 (*)

(*) i s t r u e i f f (1) 2^{u + 1} - 1 > 0 a n d 2^{u} - 2 > 0

o r (2) 2^{u + 1} - 1 < 0 a n d 2^{u} - 2 < 0 .

F r o m (1) 2^{u + 1} - 1 > 0 \Rightarrow u + 1 > 0 \Rightarrow u > - 1

a n d 2^{u} - 2 > 0 \Rightarrow u > 1 . H e n c e, u > 1 .

S i n c e u = {l o g}_{0.5}^{2} x \Rightarrow {l o g}_{0.5}^{2} x > 1

({l o g}_{0.5} x + 1) ({l o g}_{0.5} x - 1) > 0

\Rightarrow (3) {l o g}_{0.5} x + 1 > 0 a n d {l o g}_{0.5} x - 1 > 0

o r (4) {l o g}_{0.5} x + 1 < 0 a n d {l o g}_{0.5} x - 1 < 0 .

F r o m (3), x > 0 . 5^{- 1} = 2 a n d x > 0 . 5^{1} = 0.5

∴ x > 2 .

F r o m (4) 0 < x < 0 . 5^{- 1} = 2 a n d 0 < x < 0.5

∴ 0 < x < 0.5 .

F r o m (2) 2^{u + 1} - 1 < 0 \Rightarrow u + 1 < 0 \Rightarrow u < - 1

a n d 2^{u} - 2 < 0 \Rightarrow u < 1 ∴ u < - 1 . S i n c e

u = {l o g}_{0.5}^{2} x \Rightarrow {l o g}_{0.5}^{2} x < - 1 . B u t f o r x \in R,

{l o g}_{0.5}^{2} x ≮ - 1 < 0 . ∴ f o r x \in R, (2) h a s n o

s o l u t i o n s .

F o r r e a l x, t h e s o l u t i o n s e t Υ o f t h e

i n e q u a l i t y g i v e n i s Υ = {x \in R ∣ 0 < x < 1 / 2 o r x > 2} .

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