Question

Expand the log completely

Answer 1

1/3log₂(x) +1/3log₂(y) +1/3log₂(z)

Explanation:

You want the complete expansion of ...

`\log_2{\sqrt[3]{x\cdot y\cdot z\vphantom{b}}}`

Relevant rules

The rules of logs and exponents relevant to this expansion are ...

`\log_b{(xy)}=\log_b{(x)}+\log_b{(y)}\\\\\log_b{(x^a)}=a\cdot\log_b{(x)}\\\\\sqrt[n]{\vphantom{b}a}=a^{(1)/(n)}`

Expansion

Applying these rules to the given expression, we get ...

`\log_2{\sqrt[3]{x\cdot y\cdot z\vphantom{b}}}=\log_2{((xyz)^{(1)/(3)})}=(1)/(3)\log_2{(xyz)}\\\\=\boxed{(1)/(3)\log_2{(x)}+(1)/(3)\log_2{(y)}+(1)/(3)\log_2{(z)}}`