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

\newline

Answer:

Apply Quotient Rule: Apply the quotient rule of logarithms to the expression $\log\left(\frac{yx^{4}}{\sqrt[3]{z^{4}}}\right)$ . $\newline$ Quotient rule of logarithm: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ $\newline$ $\log\left(\frac{yx^{4}}{\sqrt[3]{z^{4}}}\right) = \log(yx^{4}) - \log(\sqrt[3]{z^{4}})$
Apply Product Rule: Apply the product rule of logarithms to the numerator $\log(yx^{4})$ . $\newline$ Product rule of logarithm: $\log(a \cdot b) = \log(a) + \log(b)$ $\newline$ $\log(yx^{4}) = \log(y) + \log(x^{4})$
Apply Power Rule: Apply the power rule of logarithms to $\log(x^{4})$ . $\newline$ Power rule of logarithm: $\log(a^{n}) = n \cdot \log(a)$ $\newline$ $\log(x^{4}) = 4 \cdot \log(x)$
Apply Power Rule: Apply the power rule of logarithms to the denominator $\log(\sqrt[3]{z^{4}})$ . The cube root can be written as a power of $\frac{1}{3}$ , and the power rule of logarithms can be applied. $\log(\sqrt[3]{z^{4}}) = \log((z^{4})^{\frac{1}{3}})$ $\log((z^{4})^{\frac{1}{3}}) = \frac{1}{3} \cdot \log(z^{4})$
Apply Power Rule: Apply the power rule of logarithms to $\log(z^{4})$ . $\newline$ $\log(z^{4}) = 4 \cdot \log(z)$
Substitute and Simplify: Substitute the results from Steps $3$ and $5$ into the expression from Step $1$ . $\log\left(\frac{y x^{4}}{\sqrt[3]{z^{4}}}\right) = (\log(y) + 4 \cdot \log(x)) - \left(\frac{1}{3}\right) \cdot (4 \cdot \log(z))$
Distribute and Simplify: Distribute the $\frac{1}{3}$ in the second term and simplify the expression. $\log\left(\frac{yx^{4}}{\sqrt[3]{z^{4}}}\right) = \log(y) + 4 \cdot \log(x) - \left(\frac{4}{3}\right) \cdot \log(z)$