If $l =\text{log }\dfrac{5}{7}, m=\text{log }\dfrac{7}{9}$ and $n =2(\text{log }3 - \text{log }\sqrt5)$; find the value of :

(i) $l + m + n$

(ii) $7^{l+m+n}$

Given: $l =\text{log }\dfrac{5}{7}, m=\text{log }\dfrac{7}{9}$ and $n =2(\text{log }3 - \text{log }\sqrt5)$ = 2 log $\dfrac{3}{\sqrt5}$

(i) $l + m + n$

= $\text{log }\dfrac{5}{7} + \text{log }\dfrac{7}{9} + 2 \text{log }\dfrac{3}{\sqrt5}$

= (log 5 - log 7) + (log 7 - log 9) + 2(log 3 - log $\sqrt5$)

= log 5 - log 7 + log 7 - log 9 + 2log 3 - 2log $\sqrt5$

= log 5 - log 9 + log $3^2$ - log $(\sqrt5)^2$

= log 5 - log 9 + log 9 - log 5

= 0

Hence, the value of l + m + n = 0.

(ii) $7^{l+m+n}$

From (i), we know l + m + n = 0,

⇒ $7^0$ = 1

Hence, the value of $7^{l+m+n} = 1$.