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Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log x,log y, and
log z.

log ((root(3)(x))/(zy^(2)))
Answer:

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

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Quotient property of logarithms

Full solution

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]{x}}{z y^{2}}

\newline

Answer:

Apply Quotient Rule: Apply the quotient rule of logarithms to the expression $\log\left(\frac{\sqrt[3]{x}}{zy^{2}}\right)$ . Quotient rule of logarithm: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ $\log\left(\frac{\sqrt[3]{x}}{zy^{2}}\right) = \log(\sqrt[3]{x}) - \log(zy^{2})$
Apply Cube Root Property: Apply the cube root property to the logarithm $\log(\sqrt[3]{x})$ . $\newline$ Cube root property: $\log(\sqrt[3]{a}) = \frac{1}{3} \cdot \log(a)$ $\newline$ $\log(\sqrt[3]{x}) = \frac{1}{3} \cdot \log(x)$
Apply Product Rule: Apply the product rule of logarithms to the expression $\log(zy^{2})$ . $\newline$ Product rule of logarithm: $\log(a \cdot b) = \log(a) + \log(b)$ $\newline$ $\log(zy^{2}) = \log(z) + \log(y^{2})$
Apply Power Rule: Apply the power rule to the logarithm $\log(y^{2})$ . $\newline$ Power rule of logarithm: $\log(a^{b}) = b \times \log(a)$ $\newline$ $\log(y^{2}) = 2 \times \log(y)$
Combine and Distribute: Combine the results from Steps $1$ to $4$ to get the final expanded form. $\newline$ $\log\left(\frac{\sqrt[3]{x}}{zy^{2}}\right) = \frac{1}{3} \cdot \log(x) - (\log(z) + 2 \cdot \log(y))$ $\newline$ Distribute the negative sign: $\newline$ $\log\left(\frac{\sqrt[3]{x}}{zy^{2}}\right) = \frac{1}{3} \cdot \log(x) - \log(z) - 2 \cdot \log(y)$

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