If x = log $\dfrac{3}{5}, \text{ y = log }\dfrac{5}{4}\text{ and z = 2log }\dfrac{\sqrt{3}}{2}$, find the values of

(i) x + y - z

(ii) 3^{x + y - z}

Given,

$\Rightarrow x + y - z = \text{log }\dfrac{3}{5} + \text{log }\dfrac{5}{4} - 2\text{log }\dfrac{\sqrt{3}}{2} \\[1em] = \text{log 3 - log 5 + log 5 - log 4 - 2(log }\sqrt{3} - \text{log 2)} \\[1em] = \text{log 3 - log 4 - 2log 3}^{\dfrac{1}{2}} + \text{2log 2} \\[1em] = \text{log 3 - log 2}^2 - 2 \times \dfrac{1}{2}\text{log 3 + 2log 2} \\[1em] = \text{log 3 - 2log 2 - log 3 + 2log 2} \\[1em] = 0.$

Hence, x + y - z = 0.

(ii) Given,

3^{x + y - z} = 3⁰ = 1.

Hence, 3^{x + y - z} = 1.