If p = log 20 and q = log 25, find the value of x, if 2 log (x + 1) = 2p - q.

Given,

⇒ 2 log (x + 1) = 2p - q

Substituting value of p and q in above equation, we get :

⇒ 2 log (x + 1) = 2 log 20 - log 25

⇒ log (x + 1)² = log 20² - log 25

⇒ log (x + 1)² = log 400 - log 25

⇒ log (x + 1)² = log $\dfrac{400}{25}$

⇒ log (x + 1)² = log 16

⇒ (x + 1)² = 16

⇒ x² + 1 + 2x = 16

⇒ x² + 2x + 1 - 16 = 0

⇒ x² + 2x - 15 = 0

⇒ x² + 5x - 3x - 15 = 0

⇒ x(x + 5) - 3(x + 5) = 0

⇒ (x - 3)(x + 5) = 0

⇒ x - 3 = 0 or x + 5 = 0

⇒ x = 3 or x = -5.

But x cannot be negative, then x + 1 will be negative which is not possible.

Hence, x = 3.