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{y^3 z}{\sqrt{x}}\\)

Explanation:

Step1: Use quotient rule of logs

The quotient rule of logarithms states that $\log\frac{a}{b}=\log a - \log b$. Applying this to $\log\frac{y^{3}z}{\sqrt{x}}$, we get:
$\log(y^{3}z)-\log(\sqrt{x})$

Step2: Use product rule on $\log(y^{3}z)$

The product rule of logarithms states that $\log(ab)=\log a+\log b$. Applying this to $\log(y^{3}z)$, we have:
$\log(y^{3})+\log z-\log(\sqrt{x})$

Step3: Rewrite $\sqrt{x}$ and use power rule

We know that $\sqrt{x}=x^{\frac{1}{2}}$ and the power rule of logarithms is $\log(a^{n}) = n\log a$. So, $\log(\sqrt{x})=\log(x^{\frac{1}{2}})=\frac{1}{2}\log x$ and $\log(y^{3}) = 3\log y$. Substituting these into the expression:
$3\log y+\log z-\frac{1}{2}\log x$

Answer:

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