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Question

The value of x satisfying the equation log(x+1)(2x2+7x+5)+log(2x+5)(x+1)2=4is:


A

-2

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B

2

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C

-4

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D

4

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Solution

The correct option is B

2


Explanation for the correct option:

Find the required value.

Given: log(x+1)(2x2+7x+5)+log(2x+5)(x+1)2=4

So,

log(x+1)(2x2+7x+5)+log(2x+5)(x+1)2=4⇒log(x+1)(2x2+2x+5x+5)+log(2x+5)(x+1)2=4⇒log(x+1)2xx+1+5(x+1)+log(2x+5)(x+1)2=4⇒log(x+1)2x+5(x+1)+log(2x+5)(x+1)2=4⇒log(x+1)2x+5+log(x+1)(x+1)+log(2x+5)(x+1)2=4∵log(mn)=logm+logn⇒log(x+1)2x+5+1+log(2x+5)(x+1)2=4∵logaa=1⇒log(x+1)2x+5+1+2log(2x+5)(x+1)=4∵logmn=nlogm⇒log(x+1)2x+5+2log(x+1)2x+5-3=0∵logab=1logba

Put, log(x+1)(2x+5)=t

⇒t+2t-3=0⇒t2+2-3tt=0⇒t2+2-3t=0⇒t2-2t-t+2=0⇒tt-2-1t-2=0⇒t-1t-2=0⇒t=1,2

For t=1

log(x+1)(2x+5)=1⇒(2x+5)=(x+1)1∵logab=n⇒b=an⇒x=-4

Which gives the negative base of log(-4+1)=log-3 so x=-4is not possible

Now for t=2

log(x+1)(2x+5)=2⇒(2x+5)=(x+1)2∵logab=n⇒b=an⇒2x+5=x2+2x+1⇒2x+5=x2+2x+1⇒x2=4⇒x=+2,x=-2

x=-2 gives the negative base of log(-2+1)=log-1 so x=-2is not possible

Therefore the required value of x is 2

Hence, option (B) is the correct answer.


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