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Question

Expand ${log}_{a} \sqrt[3]{x^{7} y^{8} \div \sqrt[4]{z}}$

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Solution

Consider $l o g_{a} \sqrt[3]{x^{7} y^{8} \div \sqrt[4]{z}}$

$l o g_{a} \sqrt[3]{x^{7} y^{8} \div \sqrt[4]{z}} = l o g_{a} (\sqrt[3]{\frac{x^{7} y^{8}}{\sqrt[4]{z}}})$

$= l o g_{a} {(\frac{x^{7} y^{8}}{\sqrt[4]{z}})}^{\frac{1}{3}}$

$= \frac{1}{3} l o g_{a} (\frac{x^{7} y^{8}}{\sqrt[4]{z}})$

$= \frac{1}{3} {l o g_{a} (x^{7} y^{8}) - l o g_{a} (\sqrt[4]{z})}$

$= \frac{1}{3} ⎧ ⎪ ⎨ ⎪ ⎩ l o g_{a} (x^{7} y^{8}) - l o g_{a} (z)^{\frac{1}{4}} ⎫ ⎪ ⎬ ⎪ ⎭$

$= \frac{1}{3} {l o g_{a} (x^{7} y^{8}) - \frac{1}{4} l o g_{a} (z)}$

$= \frac{1}{3} {l o g_{a} (x^{7}) + l o g_{a} (y^{8}) - \frac{1}{4} l o g_{a} (z)}$

$l o g_{a} \sqrt[3]{x^{7} y^{8} \div \sqrt[4]{z}}$ $= \frac{1}{3} {7 l o g_{a} (x) + 8 l o g_{a} (y) - \frac{1}{4} l o g_{a} (z)}$

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