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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 ((z^(4)root(3)(y))/(x))
Answer:

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

\newline

Answer:

Identify Properties: Identify the properties used to expand the logarithm. $\newline$ We will use the quotient property and the power property of logarithms to expand the given expression. $\newline$ Quotient Property: $\log_b \left(\frac{P}{Q}\right) = \log_b P - \log_b Q$ $\newline$ Power Property: $\log_b (P^k) = k \cdot \log_b P$
Apply Quotient Property: Apply the quotient property to the given logarithm. $\newline$ Using the quotient property, we can separate the numerator and the denominator of the fraction inside the logarithm. $\newline$ $\log \left(\frac{z^{4}\sqrt[3]{y}}{x}\right) = \log (z^{4}\sqrt[3]{y}) - \log (x)$
Apply Power Property $z^4$ : Apply the power property to the term $z^{4}$ in the numerator. $\newline$ The term $z^{4}$ can be expanded using the power property. $\newline$ $\log (z^{4}\sqrt[3]{y}) = 4 \cdot \log (z) + \log (\sqrt[3]{y})$
Apply Power Property $\sqrt[3]{y}$ : Apply the power property to the term $\sqrt[3]{y}$ in the numerator. $\newline$ The cube root of $y$ is equivalent to $y$ raised to the power of $\frac{1}{3}$ . $\newline$ $\log (\sqrt[3]{y}) = \log (y^{\frac{1}{3}}) = \frac{1}{3} \cdot \log (y)$
Combine Results: Combine the results from steps $3$ and $4$ . $\newline$ Now we combine the logarithms of $z^{4}$ and $\sqrt[3]{y}$ into a single expression. $\newline$ $\log (z^{4}\sqrt[3]{y}) = 4 \cdot \log (z) + \frac{1}{3} \cdot \log (y)$
Final Expanded Form: Combine all the results to get the final expanded form. $\newline$ Now we combine the results from steps $2$ , $3$ , $4$ , and $5$ to get the final expanded form of the original logarithm. $\newline$ $\log \left(\frac{z^{4}\sqrt[3]{y}}{x}\right) = 4 \cdot \log (z) + \frac{1}{3} \cdot \log (y) - \log (x)$

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