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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^(2)
Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x$ , and $\log y$ . $\newline$ $\log x^{5} y^{2}$ $\newline$ 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^{2}

\newline

Answer:

Identify Properties: Identify the properties used to expand $\log(x^5 \cdot y^2)$ . We will use the product property and the power property of logarithms to expand the given expression. Product property: $\log_b(m \cdot n) = \log_b(m) + \log_b(n)$ Power property: $\log_b(m^n) = n \cdot \log_b(m)$
Apply Product Property: Apply the product property to $\log(x^5 \cdot y^2)$ . Using the product property, we can separate the logarithm of the product into the sum of the logarithms of the individual factors. $\log(x^5 \cdot y^2) = \log(x^5) + \log(y^2)$
Apply Power Property: Apply the power property to each logarithm. $\newline$ Now we apply the power property to both $\log(x^5)$ and $\log(y^2)$ to bring the exponents out in front of the logarithms. $\newline$ $\log(x^5) = 5 \cdot \log(x)$ $\newline$ $\log(y^2) = 2 \cdot \log(y)$
Combine Results: Combine the results from Step $3$ to get the final expanded form. $\newline$ Combining the results from Step $3$ , we get: $\newline$ $\log(x^5 \cdot y^2) = 5 \cdot \log(x) + 2 \cdot \log(y)$

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