Question

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 \dfrac{z^2\sqrt3{x^4}}{y}\\)

Explanation:

Step1: Apply Quotient Rule

The quotient rule of logarithms states that $\log \frac{a}{b} = \log a - \log b$. So, for $\log \frac{z^{2}\sqrt[3]{x^{4}}}{y}$, we can split it as $\log (z^{2}\sqrt[3]{x^{4}}) - \log y$.

Step2: Apply Product Rule

The product rule of logarithms states that $\log (ab) = \log a + \log b$. So, $\log (z^{2}\sqrt[3]{x^{4}}) = \log z^{2} + \log \sqrt[3]{x^{4}}$.

Step3: Simplify $\log z^{2}$

Using the power rule of logarithms, $\log a^{n} = n\log a$. So, $\log z^{2} = 2\log z$.

Step4: Simplify $\log \sqrt[3]{x^{4}}$

First, rewrite $\sqrt[3]{x^{4}}$ as $x^{\frac{4}{3}}$. Then, using the power rule, $\log x^{\frac{4}{3}} = \frac{4}{3}\log x$.

Step5: Combine All Terms

Putting it all together: $\log (z^{2}\sqrt[3]{x^{4}}) - \log y = 2\log z + \frac{4}{3}\log x - \log y$.

Answer:

$2\log z + \frac{4}{3}\log x - \log y$