Question

1. \log \sqrt3{x \cdot y \cdot z}

Explanation:

Step1: Convert radical to exponent

The cube root \(\sqrt[3]{x\cdot y\cdot z}\) can be written as \((x\cdot y\cdot z)^{\frac{1}{3}}\) using the property \(\sqrt[n]{a}=a^{\frac{1}{n}}\). So the expression becomes \(\log((x\cdot y\cdot z)^{\frac{1}{3}})\).

Step2: Apply logarithm power rule

Using the power rule of logarithms \(\log(a^b)=b\log(a)\), we get \(\frac{1}{3}\log(x\cdot y\cdot z)\).

Step3: Apply logarithm product rule

Using the product rule of logarithms \(\log(a\cdot b\cdot c)=\log(a)+\log(b)+\log(c)\), we can expand \(\log(x\cdot y\cdot z)\) to \(\log(x)+\log(y)+\log(z)\). Then multiply by \(\frac{1}{3}\): \(\frac{1}{3}\log(x)+\frac{1}{3}\log(y)+\frac{1}{3}\log(z)\).

Answer:

\(\frac{1}{3}\log x + \frac{1}{3}\log y + \frac{1}{3}\log z\)