Express in terms of log 2 and log 3 :

(i) log $\dfrac{\sqrt8}{27}$

(ii) log $(\sqrt{54}\times \sqrt[3]{243})$

(i) log $\dfrac{\sqrt8}{27}$

= log $\sqrt8$ - log 27

= log $8^{\dfrac{1}{2}}$ - log $3^3$

= log $(2^3)^{\dfrac{1}{2}}$ - 3log 3

= log $(2)^{\dfrac{3}{2}}$ - 3log 3

= $\dfrac{3}{2}$ log 2 - 3log 3

Hence, the value of log $\dfrac{\sqrt8}{27} = \dfrac{3}{2}$ log 2 - 3log 3.

(ii) log $(\sqrt{54}\times \sqrt[3]{243})$

= log $54^{\dfrac{1}{2}}$ + log $243^{\dfrac{1}{3}}$

= log $(2 \times 27)^{\dfrac{1}{2}}$ + log $(3 ^5)^{\dfrac{1}{3}}$

= log $(2 \times 3^3)^{\dfrac{1}{2}}$ + log $3^{\dfrac{5}{3}}$

= log $(2)^{\dfrac{1}{2}}$ + log $(3^3)^{\dfrac{1}{2}}$ + log $(3)^{\dfrac{5}{3}}$

= log $(2)^{\dfrac{1}{2}}$ + log $(3)^{\dfrac{3}{2}}$ + log $(3)^{\dfrac{5}{3}}$

= ${\dfrac{1}{2}}$ log 2 + ${\dfrac{3}{2}}$ log 3 + ${\dfrac{5}{3}}$ log 3

= ${\dfrac{1}{2}}$ log 2 + $\Big({\dfrac{3}{2}} + {\dfrac{5}{3}}\Big)$ log 3

= ${\dfrac{1}{2}}$ log 2 + $\Big({\dfrac{9 + 10}{6}}\Big)$ log 3

= ${\dfrac{1}{2}}$ log 2 + ${\dfrac{19}{6}}$ log 3

Hence, the value of log $(\sqrt{54}\times \sqrt[3]{243}) = {\dfrac{1}{2}}$ log 2 + ${\dfrac{19}{6}}$ log 3.