The product of all positive real values of x satisfying the equation x 16 log 5 x 3 - 68 log 5 x = 5 - 16 is

Step-by-step solution

Taking log to the base 5 on both sides 16 log 5 x 3 - 68 log 5 x log 5 x = - 16 Let log 5 x = t 16 t 4 - 68 t 2 + 16 = 0 ⇒ 4 t 4 - 16 t 2 - t 2 + 4 = 0 ⇒ 4 t 2 - 1 t 2 - 4 = 0 ⇒ t = ± 1 2 , ± 2 So log 5 x = ± 1 2 or ± 2 ⇒ x = 5 1 2 ,   5 - 1 2 ,   5 2 ,   5 - 2 Hence, the product is 5 × 1 5 × 25 × 1 25 = 1