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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))
Answer:

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

\newline

Answer:

Identify Properties: Identify the properties used to expand $\log\left(\frac{x^{2}}{y}\right)$ . We have a logarithm of a quotient and a power. We will use the quotient property and the power property of logarithms to expand this expression. 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 $\log\left(\frac{x^{2}}{y}\right)$ . Using the quotient property, we can write $\log\left(\frac{x^{2}}{y}\right)$ as $\log(x^{2}) - \log(y)$ .
Apply Power Property: Apply the power property to the logarithm $\log(x^{2})$ . Using the power property, we can write $\log(x^{2})$ as $2 \times \log(x)$ .
Combine Results: Combine the results from Step $2$ and Step $3$ to get the final expanded form. $\newline$ The final expanded form is $2 \cdot \log(x) - \log(y)$ .

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