If x = 1 + log 2 - log 5, y = 2 log 3 and z = log a - log 5; find the value of a, if x + y = 2z.

Given,

⇒ x + y = 2z

⇒ 1 + log 2 - log 5 + 2 log 3 = 2 (log a - log 5)

⇒ log 10 + log 2 - log 5 + log 3² = 2 log $\dfrac{a}{5}$

⇒ log 10 + log 2 + log 9 - log 5 = log $\Big(\dfrac{a}{5}\Big)^2$

⇒ log $\dfrac{10 \times 2 \times 9}{5}$ = log $\Big(\dfrac{a}{5}\Big)^2$

⇒ 2 × 2 × 9 = $\Big(\dfrac{a}{5}\Big)^2$

⇒ a² = 5² × 2 × 2 × 9

⇒ a² = 900

⇒ a = $\sqrt{900}$ = 30.

Hence, a = 30.