GATE2012 AR: GA-6

Question

A value of $x$ that satisfies the equation $\log x + \log (x – 7) = \log (x + 11) + \log 2$ is

• A. $1$ 
• B. $2$ 
• C. $7$ 
• D. $11$

Comments

Dear Admin, can this be categorized under logarithms?

Answer 1

$\log m + \log n = \log mn$

So, $\log x + \log (x-7) = \log (x+11) + \log 2$

$\Rightarrow \log x(x-7) = \log 2 (x+11)$

$\Rightarrow x(x-7) = 2 (x+11)$

$\Rightarrow x^2-9x-22=0$

$\Rightarrow (x-11)(x+2)=0$

$\therefore x=11$

$\because x \neq -2$ , log is undefined for negative number.

Correct Answer: $D$

Comments

sir your approach is right but just knowing basic of log without solving one can get an answer

1. log value always positive

1. log value can't be zero (bcoz at zero it's undefined)

so. given  log(x-7)      x-7>0    x>7  // only one value is in option greater than 7 which is 11

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You are right

Answer 2

Another approach try substitution of all the answers 1,2,7 doesn't satisfy the equation as value of log is never negative and zero.

So answer is 11 so lets substitute 11 and verify.

log 11+log 4=log(22)+log 2

log(22)=log(11*2)=log(11)+log(2)

log 11+ log 4=log 11+ log 2+ log 2

log 11+ log 4= log 11 + 2 log 2

log 11 + 2 log 2= log 11 + 2 log 2

LHS = RHS