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{\sqrt{z^5}x}{y^2}\\)

Explanation:

Step1: Use quotient rule of logs

The quotient rule of logarithms states that $\log \frac{a}{b} = \log a - \log b$. So, we can rewrite $\log \frac{\sqrt{z^5}x}{y^2}$ as $\log (\sqrt{z^5}x) - \log (y^2)$.

Step2: Use product rule of logs

The product rule of logarithms states that $\log (ab) = \log a + \log b$. So, we can rewrite $\log (\sqrt{z^5}x)$ as $\log (\sqrt{z^5}) + \log x$.

Step3: Simplify $\log (\sqrt{z^5})$

We know that $\sqrt{z^5} = z^{\frac{5}{2}}$, so $\log (\sqrt{z^5}) = \log (z^{\frac{5}{2}})$.

Step4: Use power rule of logs

The power rule of logarithms states that $\log (a^n) = n\log a$. So, $\log (z^{\frac{5}{2}}) = \frac{5}{2}\log z$ and $\log (y^2) = 2\log y$.

Step5: Combine all the terms

Putting it all together: $\log (\sqrt{z^5}) + \log x - \log (y^2) = \frac{5}{2}\log z + \log x - 2\log y$.

Answer:

$\frac{5}{2}\log z + \log x - 2\log y$