Expand $\text{log}_a \space {\sqrt[3]{x^7y^8 ÷ \sqrt[4]{z}}}$

Given,

$\Rightarrow \text{log}$a \space {\sqrt[3]{x^7y^8 ÷ \sqrt[4]{z}}} \\[1em] \Rightarrow \text{log}a \space {(x^7y^8 ÷ \sqrt[4]{z})^{\dfrac{1}{3}}} \\[1em] \Rightarrow \dfrac{1}{3}\text{log}a \space {(x^7y^8 ÷ \sqrt[4]{z})} \\[1em] \Rightarrow \dfrac{1}{3}[\text{log}a \space x^7y^8 - \text{log}a \space \sqrt[4]{z}] \\[1em] \Rightarrow \dfrac{1}{3}[\text{log}a \space x^7 + \text{log}a \space y^8 - \text{log}a \space z^{\dfrac{1}{4}}] \\[1em] \Rightarrow \dfrac{1}{3}[7\text{log}a \space x + 8\text{log}a \space y - \dfrac{1}{4}\text{log}a \space z] \\[1em] \Rightarrow \dfrac{7}{3}\text{log}a \space x + \dfrac{8}{3}\text{log}a \space y - \dfrac{1}{12}\text{log}a \space z.

Hence, $\text{log}$a \space {\sqrt[3]{x^7y^8 ÷ \sqrt[4]{z}}} = \dfrac{7}{3}\text{log}a \space x + \dfrac{8}{3}\text{log}a \space y - \dfrac{1}{12}\text{log}a \space z.