If α and β are the roots of the equation ( log 2 x)2+4( log 2 x)-1=0 then the value of log β α+ log α β equals

Question

If α and β are the roots of the equation ( log 2 x)2+4( log 2 x)-1=0 then the value of log β α+ log α β equals (A) 18 (B) -16 (C) 14 (D) -18

Tardigrade Admin · Accepted Answer

log 2 α+ log 2 β=-4 ; log 2 α ⋅ log 2 β=-1 now log β α+ log α β=( log 2 α/ log 2 β)+( log 2 β/ log 2 α)=(( log 2 α)2+( log 2 β)2/ log 2 α ⋅ log 2 β) =-[( log 2 α+ log 2 β)2-2 log 2 α ⋅ log 2 β] =-[16+2]=-18.

Q. If α and β are the roots of the equation (log2​x)2+4(log2​x)−1=0 then the value of logβ​α+logα​β equals

18

-16

14

-18

log2​α+log2​β=−4;log2​α⋅log2​β=−1 now logβ​α+logα​β=log2​βlog2​α​+log2​αlog2​β​=log2​α⋅log2​β(log2​α)2+(log2​β)2​ =−[(log2​α+log2​β)2−2log2​α⋅log2​β] =−[16+2]=−18.

Q. If $α$ and $β$ are the roots of the equation $(lo g_{2} x)^{2} + 4 (lo g_{2} x) - 1 = 0$ then the value of $lo g_{β} α + lo g_{α} β$ equals