If x1, x2(x2x1) satisfy the equation, 5 log x / 9(x)+ log 9 / x(x3)+8 log 9 x2(x2)=2 then

Question

If x1, x2(x2>x1) satisfy the equation, 5 log x / 9(x)+ log 9 / x(x3)+8 log 9 x2(x2)=2 then (A)  log 3( x 1 x 2)=( log 3 x 1)( log 3 x 2) (B) x 1 x 2= x 1 x 2 (C) x 22- x 12=3 log x 1( x 2) (D) the sum of the roots of the equation is 3 / 2.

Tardigrade Admin · Accepted Answer

(5 log 3 x/ log 3 x-2)+(3 log 3 x/2- log 3 x)+(16 log 3 x/2+2 log 3 x)=2 text let log 3 x = t (5 t / t -2)-(3 t / t -2)+(8 t /1+ t )=2 ⇒ (5 t -3 t / t -2)+(8 t /1+ t )=2 ⇒ (2 t / t -2)+(8 t /1+ t )=2 ⇒ 4 t2-6 t+2=0 ⇒ 2 t2-3 t+1=0

Q. If $x_{1}, x_{2} (x_{2} > x_{1})$ satisfy the equation, $5 lo g_{x /9} (x) + lo g_{9/ x} (x^{3}) + 8 lo g_{9 x^{2}} (x^{2}) = 2$ then

$lo g_{3} (x_{1} x_{2}) = (lo g_{3} x_{1}) (lo g_{3} x_{2})$

$x_{1}^{x_{2}} = x_{1} x_{2}$

$x_{2}^{2} - x_{1}^{2} = 3 lo g_{x_{1}} (x_{2})$

the sum of the roots of the equation is $3/2$ .

Q. If x1​,x2​(x2​>x1​) satisfy the equation, 5logx/9​(x)+log9/x​(x3)+8log9x2​(x2)=2 then

log3​(x1​x2​)=(log3​x1​)(log3​x2​)

x1x2​​=x1​x2​

x22​−x12​=3logx1​​(x2​)

the sum of the roots of the equation is 3/2.

log3​x−25log3​x​+2−log3​x3log3​x​+2+2log3​x16log3​x​=2 let log3​x=t t−25t​−t−23t​+1+t8t​=2 ⇒t−25t−3t​+1+t8t​=2 ⇒t−22t​+1+t8t​=2 ⇒4t2−6t+2=0 ⇒2t2−3t+1=0

Q. If $x_{1}, x_{2} (x_{2} > x_{1})$ satisfy the equation, $5 lo g_{x /9} (x) + lo g_{9/ x} (x^{3}) + 8 lo g_{9 x^{2}} (x^{2}) = 2$ then

$lo g_{3} (x_{1} x_{2}) = (lo g_{3} x_{1}) (lo g_{3} x_{2})$

$x_{1}^{x_{2}} = x_{1} x_{2}$

$x_{2}^{2} - x_{1}^{2} = 3 lo g_{x_{1}} (x_{2})$

the sum of the roots of the equation is $3/2$ .