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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^(2))/(y^(4)))
Answer:

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

\newline

Answer:

Identify Properties: Identify the properties used to expand $\log\left(\frac{x^{2}}{y^{4}}\right)$ . We will use the quotient property of logarithms to separate the numerator and denominator, and the power property to bring down the exponents. 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 $\log\left(\frac{x^{2}}{y^{4}}\right)$ as the difference of two logarithms: $\newline$ $\log\left(\frac{x^{2}}{y^{4}}\right) = \log(x^{2}) - \log(y^{4})$
Apply Power Property: Apply the power property to both logarithms. $\newline$ Now we apply the power property to bring down the exponents: $\newline$ $\log(x^{2})$ becomes $2 \cdot \log(x)$ , and $\log(y^{4})$ becomes $4 \cdot \log(y)$ . $\newline$ So, $\log\left(\frac{x^{2}}{y^{4}}\right) = 2 \cdot \log(x) - 4 \cdot \log(y)$

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