If log 3 2 , log 3 2 x - 5 , log 3 2 x - 7 2 are in an arithmetic progression, then the value of x is equal to _____.

Step-by-step solution

As log 3 2 , log 3 2 x - 5 , log 3 2 x - 7 2 are in arithmetic progression, 2 log 3 2 x - 5 = log 3 2 + log 3 2 x - 7 2 ⇒ log 3 2 x - 5 2 = log 3 2 2 x - 7 2 Let 2 x = t So, log 3 t - 5 2 = log 3 2 t - 7 2 ⇒ t - 5 2 = 2 t - 7 ⇒ t 2 - 12 t + 32 = 0 ⇒ t - 4 t - 8 = 0 ⇒ 2 x = 4 or 2 x = 8 Here, x ≠ 2 , as log 3 4 - 5 does not exist. So, x = 3