log-2x (6x2 -5x + 1) – log-3x(4x2 - 4x + 1) = 2.,
Answers
Answer:
log
3x+7
(4x
2
+12x+9)+log
2x+3
(6x
2
+23x+12)=4
Consider 4x
2
+12x+9=(2x)
2
+2×2x×3+3
2
=(2x+3)
2
And 6x
2
+23x+21=0
⇒6x
2
+9x+14x+21=0
⇒3x(2x+3)+7(2x+3)=0
⇒(3x+7)(2x+3)=0
⇒log
3x+7
(2x+3)
2
+log
2x+3
(3x+7)(2x+3)=4
⇒2log
(3x+7)
(2x+3)+log
(2x+3)
(3x+7)+log
(2x+3)
(2x+3)=4 since log
a
a=1
⇒2log
(3x+7)
(2x+3)+log
(2x+3)
(3x+7)+1=4
⇒2log
(3x+7)
(2x+3)+log
(2x+3)
(3x+7)=3
Let α=log
(3x+7)
(2x+3) then log
(2x+3)
(3x+7)=
α
1
⇒2α+
α
1
=3
⇒2α
2
−3α+1=0 is quadratic in α
⇒2α
2
−2α−α+1=0
⇒2α(α−1)−1(α−1)=0
⇒(α−1)(2α−1)=0
⇒α=1,α=
2
1
⇒log
(3x+7)
(2x+3)=1,
2
1
where α=log
(3x+7)
(2x+3)
⇒2x+3=3x+7,2x+3=(3x+7)
2
1
⇒2x−3x=7−3,(2x+3)
2
=3x+7
⇒x=−4,4x
2
+9+12x−3x−7=0
⇒x=−4,4x
2
+9x+2=0
⇒x=−4,4x
2
+8x+x+2=0
⇒x=−4,4x(x+2)+(x+2)=0
⇒x=−4.(4x+1)=0,(x+2)=0
⇒x=−4,x=
4
−1
,−2
For x=−2,−4 the base will become negative.
∴x=
4
−1
Step-by-step explanation:
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