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x-log-2-x-4-lt-x-16-




Question Number 140589 by bramlexs22 last updated on 09/May/21
 x^(−log _2  x +4)  < (x/(16))
xlog2x+4<x16
Answered by MJS_new last updated on 10/May/21
x^(4−log_2  x) ∈R ⇒ x>0  let x=2^t   (2^t )^(4−t) <(2^t /2^4 )  2^(4t−t^2 ) <2^(t−4)   4t−t^2 <t−4  t^2 −3t−4>0  ⇒ t<−1∨t>4  ⇒ x<(1/2)∨x>16  ⇒  0<x<(1/2)∨x>16
x4log2xRx>0letx=2t(2t)4t<2t2424tt2<2t44tt2<t4t23t4>0t<1t>4x<12x>160<x<12x>16
Commented by bramlexs22 last updated on 10/May/21
nice sir
nicesir
Answered by bemath last updated on 10/May/21
log _2 (x) ( 4−log _2 (x))< log _2 (x)−4  let log _2 (x)= b  ⇒4b−b^2  < b−4   ⇒b^2 −3b−4 > 0  ⇒(b−4)(b+1) >0  ⇒ b<−1 ∨ b > 4  ⇒ 0 < x < (1/2) ∨ x > 16
log2(x)(4log2(x))<log2(x)4letlog2(x)=b4bb2<b4b23b4>0(b4)(b+1)>0b<1b>40<x<12x>16

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