If the reciprocals of 2 , log 3 x - 4 ⁡ 4 and log 3 x + 7 2 ⁡ 4 are in arithmetic progression, then x is equal to

Step-by-step solution

Reciprocals of 2 , log ⁡ 4 log ⁡ 3 x - 4 , log ⁡ 4 log ⁡ 3 x + 7 2 are in A.P. ⇒ 1 2 , log ⁡ 3 x - 4 log ⁡ 4 , log ⁡ 3 x + 7 2 log ⁡ 4 are in A.P. ⇒ 1 2 log ⁡ 4 , log ⁡ 3 x - 4 , log ⁡ 3 x + 7 2 are in A.P. ⇒ log ⁡ 2 , log ⁡ 3 x - 4 , log ⁡ 3 x + 7 2 are in A.P. ⇒ 2 , 3 x - 4 , 3 x + 7 2 are in G.P. ⇒ 3 x - 4 2 = 2 3 x + 7 2 Let 3 x = y ⇒ y - 4 2 = 2 y + 7 2 ⇒ y 2 + 16 - 8 y = 2 y + 7 ⇒ y 2 - 10 y + 9 = 0 ⇒ y - 1 y - 9 = 0 ⇒ y = 1,9 ⇒ 3 x = 1,9 ⇒ x = 0,2 But, for x = 0 , 3 x - 4 = - 3 < 0 So, x = 2 will only form a