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\sqrt{z}}{x}\\)

Explanation:

Step1: Apply Quotient Rule

The quotient rule for logarithms states that $\log \frac{a}{b} = \log a - \log b$. So, for $\log \frac{y\sqrt{z}}{x}$, we can split it as $\log(y\sqrt{z}) - \log x$.

Step2: Apply Product Rule to $\log(y\sqrt{z})$

The product rule for logarithms is $\log(ab) = \log a + \log b$. Here, $a = y$ and $b = \sqrt{z}$, so $\log(y\sqrt{z}) = \log y + \log \sqrt{z}$.

Step3: Simplify $\log \sqrt{z}$

Recall that $\sqrt{z} = z^{\frac{1}{2}}$. Using the power rule for logarithms, $\log a^n = n\log a$, we get $\log z^{\frac{1}{2}} = \frac{1}{2}\log z$.

Step4: Combine All Terms

Substituting back, we have $\log y + \frac{1}{2}\log z - \log x$.

Answer:

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