If p = log₁₀ 20 and q = log₁₀ 25, find the value of x if

2log₁₀ (x + 1) = 2p - q

Given,

2log₁₀(x + 1) = 2p - q

⇒ 2log₁₀(x + 1) = 2log₁₀20 - log₁₀25

⇒ log₁₀(x + 1)² = log₁₀20² - log₁₀25

⇒ log₁₀(x + 1)² = log₁₀400 - log₁₀25

⇒ log₁₀(x + 1)² = $\text{log}_{10}\space\dfrac{400}{25}$

⇒ log₁₀(x + 1)² = log₁₀16

⇒ (x + 1)² = 16

⇒ x² + 1 + 2x = 16

x² + 2x - 15 = 0

x² + 5x - 3x - 15 = 0

x(x + 5) - 3(x + 5) = 0

(x - 3)(x + 5) = 0

x = 3 or -5.

But x ≠ -5 as then (x + 1) will be negative.

Hence, x = 3.