Given :

$\dfrac{\text{log x}}{\text{log y}} = \dfrac{3}{2}$ and log (xy) = 5; find the values of x and y.

Solving, equation :

$\Rightarrow \dfrac{\text{log x}}{\text{log y}} = \dfrac{3}{2} \\[1em] \Rightarrow \text{2 log x = 3 log y} \\[1em] \Rightarrow \text{log } x^2 = \text{log } y^3 \\[1em] \Rightarrow x^2 = y^3 \\[1em] \Rightarrow x = \sqrt{y^3}\text{ ……..(1)}$

Substituting value of x from equation (1) in log (xy) = 5, we get :

$\Rightarrow \text{log } (\sqrt{y^3} \times y) = 5 \\[1em] \Rightarrow \text{log } (y^{\dfrac{3}{2}}\times y) = 5 \\[1em] \Rightarrow \text{log } y^{\dfrac{3}{2} + 1} = 5 \\[1em] \Rightarrow y^{\dfrac{5}{2}} = 10^5 \\[1em] \Rightarrow (y^{\dfrac{1}{2}})^5 =10^5 \\[1em] \Rightarrow (\sqrt{y})^5 = 10^5 \\[1em] \Rightarrow \sqrt{y} = 10$

Squaring both sides, we get :

$\Rightarrow y = 10^2$ = 100.

Substituting value of y in equation (1), we get :

$\Rightarrow x = \sqrt{(10^2)^3} \\[1em] = \sqrt{10^6} \\[1em] = 10^3 \\[1em] = 1000.$

Hence, x = 1000 and y = 100.