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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 $\log y$ . $\newline$ $\log x^{2} y^{4}$ $\newline$ 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^{2} y^{4}

\newline

Answer:

Identify Properties: Identify the properties used to expand $\log(x^{2}y^{4})$ . We will use the product property of logarithms to separate the terms and the power property to bring down the exponents. 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 $\log(x^{2}y^{4})$ . Using the product property, we can write $\log(x^{2}y^{4})$ as the sum of two logs: $\log(x^{2}) + \log(y^{4})$ . $\log(x^{2}y^{4}) = \log(x^{2}) + \log(y^{4})$
Apply Power Property: Apply the power property to both $\log(x^{2})$ and $\log(y^{4})$ . Using the power property, we can bring the exponents out in front of the logs: $\log(x^{2}) = 2 \times \log(x)$ $\log(y^{4}) = 4 \times \log(y)$
Combine Results: Combine the results from Step $3$ to get the final expanded form. $\newline$ The final expanded form of the logarithm is the sum of the results from Step $3$ : $\newline$ $\log(x^{2}y^{4}) = 2 \cdot \log(x) + 4 \cdot \log(y)$

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