Solve for x and y:

$\dfrac{\text{log} \space x}{3} = \dfrac{\text{log} \space y}{2}$ and $\text{log} \space (xy) = 5$.

Given,

$\dfrac{\text{log} \space x}{3} = \dfrac{\text{log} \space y}{2}$

⇒ 2 log x = 3 log y

⇒ 2 log x - 3 log y = 0 …….(i)

Given,

log (xy) = 5

⇒ log x + log y = 5 ……(ii)

Multiplying (ii) by 2 we get,

⇒ 2 log x + 2 log y = 10 ……(iii)

Subtracting (i) from (iii) we get,

⇒ 2 log x + 2 log y - (2 log x - 3 log y) = 10 - 0

⇒ 2 log x - 2 log x + 2 log y + 3 log y = 10

⇒ 5 log y = 10

⇒ log y = 2

⇒ y = 10² = 100.

Substituting value of y in (ii) we get,

⇒ log x + log 100 = 5

⇒ log x + 2 = 5

⇒ log x = 3

⇒ x = 10³ = 1000.

Hence, x = 1000 and y = 100.