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

log ((x^(3))/(y^(2)))
Answer:

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

, and

\log y

.

\newline

\log \frac{x^{3}}{y^{2}}

\newline

Answer:

Identify Properties: Identify the properties used to expand $\log\left(\frac{x^{3}}{y^{2}}\right)$ . We will use the quotient property of logarithms to separate the numerator and denominator, and the power property to bring the exponents out in front of the logs. Quotient Property: $\log_b\left(\frac{P}{Q}\right) = \log_b(P) - \log_b(Q)$ Power Property: $\log_b(P^k) = k \cdot \log_b(P)$
Apply Quotient Property: Apply the quotient property to the logarithm. $\newline$ Using the quotient property, we can write $\log\left(\frac{x^{3}}{y^{2}}\right)$ as $\log(x^{3}) - \log(y^{2})$ .
Apply Power Property: Apply the power property to both logarithms. $\newline$ Using the power property, we can bring the exponents out in front of each log: $\newline$ $\log(x^{3})$ becomes $3 \times \log(x)$ , and $\log(y^{2})$ becomes $2 \times \log(y)$ .
Write Final Form: Write the final expanded form of the logarithm. $\newline$ Combining the results from steps $2$ and $3$ , we get: $\newline$ $3 \cdot \log(x) - 2 \cdot \log(y)$

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