The solution x of the equation log 4(3 x+7)- log 4(x-5)=2 would lie within which of the given ranges?

Question

The solution x of the equation log 4(3 x+7)- log 4(x-5)=2 would lie within which of the given ranges? (A) 0 ≤ x ≤ 3 (B) 3 ≤ x ≤ 6 (C) 6 ≤ x ≤ 9 (D) 9 ≤ x ≤ 12

Tardigrade Admin · Accepted Answer

log 4(3 x+7)- log 4(x-5)=2 ⇒ ((3 x+7)/(x-5))=42 ⇒ ((3 x+7)/(x-5))=16 ⇒ 3 x+7=16 x-80 ⇒ x=(87/13)>6

Q. The solution $x$ of the equation $lo g_{4} (3 x + 7) - lo g_{4} (x - 5) = 2$ would lie within which of the given ranges?

$0 \leq x \leq 3$

$3 \leq x \leq 6$

$6 \leq x \leq 9$

$9 \leq x \leq 12$

$lo g_{4} (3 x + 7) - lo g_{4} (x - 5) = 2 \Rightarrow \frac{( 3 x + 7 )}{( x - 5 )} = 4^{2} \Rightarrow \frac{( 3 x + 7 )}{( x - 5 )} = 16$
$\Rightarrow 3 x + 7 = 16 x - 80 \Rightarrow x = \frac{87}{13} > 6$

Q. The solution x of the equation log4​(3x+7)−log4​(x−5)=2 would lie within which of the given ranges?

0≤x≤3

3≤x≤6

6≤x≤9

9≤x≤12