If a = log$\dfrac{2}{3}$, b = log$\dfrac{3}{5}$ and c = 2log$\sqrt{\dfrac{5}{2}}$, find the value of

(i) a + b + c

(ii) 5^{a + b + c}

(i) Given,

$\Rightarrow a + b + c = \text{log }\dfrac{2}{3} + \text{log }\dfrac{3}{5} + 2\text{log }\sqrt{\dfrac{5}{2}} \\[1em] = \text{log 2 - log 3 + log 3 - log 5} + 2 \times \text{log}\Big(\dfrac{5}{2}\Big)^{\dfrac{1}{2}} \\[1em] = \text{log } 2 - \text{log } 5 + 2 \times \dfrac{1}{2} \times \text{(log 5 - log 2)} \\[1em] = \text{log 2 - log 5 + log 5 - log 2} \\[1em] = 0.$

Hence, a + b + c = 0.

(ii) Given,

⇒ 5^{a + b + c} = 5⁰ = 1.

Hence, 5^{a + b + c} = 1.