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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^(5)y
Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x$ , and $\log y$ . $\newline$ $\log x^{5} y$ $\newline$ Answer:

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Power 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 x^{5} y

\newline

Answer:

Identify Properties: Identify the properties used to expand $\log x^{5}y$ . We will use the product property of logarithms to separate the terms, and the power property to bring down the exponent. Product property: $\log_b(mn) = \log_b(m) + \log_b(n)$ Power property: $\log_b(m^n) = n \cdot \log_b(m)$
Apply Product Property: Apply the product property to $\log x^{5}y$ . Using the product property, we can write $\log x^{5}y$ as the sum of two logarithms: $\log x^{5} + \log y$ . $\log x^{5}y = \log x^{5} + \log y$
Apply Power Property: Apply the power property to $\log x^{5}$ . Using the power property, we can bring the exponent outside the logarithm: $5 \cdot \log x$ . $\log x^{5} = 5 \cdot \log x$
Combine Results: Combine the results from Step $2$ and Step $3$ . $\newline$ We have already separated $\log x^{5}y$ into $\log x^{5} + \log y$ and expanded $\log x^{5}$ into $5 \times \log x$ . Now we combine them to get the final expanded form. $\newline$ $\log x^{5}y = 5 \times \log x + \log y$

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