Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x, \log y$ , and $\log z$ . $\newline$ $\log \frac{\sqrt[3]{z^{4}} y}{x^{2}}$ $\newline$ Answer:

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Quotient property of logarithms

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Q. Expand the logarithm fully using the properties of logs. Express the final answer in terms of

\log x, \log y

, and

\log z

\newline

\log \frac{\sqrt[3]{z^{4}} y}{x^{2}}

\newline

Answer:

Apply Quotient Rule: Apply the quotient rule of logarithms to the expression $\log\left(\frac{\sqrt[3]{z^{4}}y}{x^{2}}\right)$ . Quotient rule of logarithm: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ $\log\left(\frac{\sqrt[3]{z^{4}}y}{x^{2}}\right) = \log(\sqrt[3]{z^{4}}y) - \log(x^{2})$
Apply Product Rule: Apply the product rule of logarithms to the numerator $\log(\sqrt[3]{z^{4}}y)$ . $\newline$ Product rule of logarithm: $\log(a \cdot b) = \log(a) + \log(b)$ $\newline$ $\log(\sqrt[3]{z^{4}}y) = \log(\sqrt[3]{z^{4}}) + \log(y)$
Rewrite Cube Root: Rewrite the cube root and the power inside the logarithm using the power rule of logarithms. $\newline$ Power rule of logarithm: $\log(a^{n}) = n \cdot \log(a)$ $\newline$ $\log(\sqrt[3]{z^{4}})$ can be written as $\log((z^{4})^{\frac{1}{3}})$ which simplifies to $(\frac{1}{3}) \cdot \log(z^{4})$ $\newline$ Now apply the power rule to $\log(z^{4})$ : $(\frac{1}{3}) \cdot 4 \cdot \log(z)$
Apply Power Rule: Apply the power rule to the denominator $\log(x^{2})$ . $\newline$ $\log(x^{2}) = 2 \cdot \log(x)$
Combine Results: Combine the results from Steps $1$ to $4$ to get the final expanded form. $\newline$ $\log\left(\frac{\sqrt[3]{z^{4}}y}{x^{2}}\right) = \left(\log(\sqrt[3]{z^{4}}) + \log(y)\right) - \log(x^{2})$ $\newline$ Substitute the expressions from Steps $3$ and $4$ : $\newline$ $\log\left(\frac{\sqrt[3]{z^{4}}y}{x^{2}}\right) = \left(\frac{1}{3} \cdot 4 \cdot \log(z) + \log(y)\right) - 2 \cdot \log(x)$
Simplify Expression: Simplify the expression. $\log\left(\frac{\sqrt[3]{z^{4}}y}{x^{2}}\right) = \frac{4}{3} \cdot \log(z) + \log(y) - 2 \cdot \log(x)$