Question

expand the logarithm fully using the properties of logs. express the final answer in terms of log $x$, log $y$, and log $z$.
$log \frac{z}{sqrt{x^3}y}$

Explanation:

Step1: Rewrite root as exponent

$\log \frac{z}{(x^3 y)^{\frac{1}{2}}}$

Step2: Apply log quotient rule

$\log z - \log(x^3 y)^{\frac{1}{2}}$

Step3: Apply log power rule

$\log z - \frac{1}{2}\log(x^3 y)$

Step4: Apply log product rule

$\log z - \frac{1}{2}(\log x^3 + \log y)$

Step5: Apply log power rule to $\log x^3$

$\log z - \frac{1}{2}(3\log x + \log y)$

Step6: Distribute the $-\frac{1}{2}$ term

$\log z - \frac{3}{2}\log x - \frac{1}{2}\log y$

Answer:

$\log z - \frac{3}{2}\log x - \frac{1}{2}\log y$