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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^(3)y^(4)
Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x$ , and $\log y$ . $\newline$ $\log x^{3} y^{4}$ $\newline$ 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^{3} y^{4}

\newline

Answer:

Identify Properties: Identify the properties used to expand $\log x^{3}y^{4}$ . $\newline$ We will use the product property of logarithms to separate the terms and the power property to bring down the exponents. $\newline$ Product property: $\log_b(mn) = \log_b(m) + \log_b(n)$ $\newline$ Power property: $\log_b(m^n) = n \cdot \log_b(m)$
Apply Product Property: Apply the product property to $\log x^{3}y^{4}$ . Using the product property, we can write $\log x^{3}y^{4}$ as the sum of two logarithms: $\log x^{3} + \log y^{4}$ . $\log x^{3}y^{4} = \log x^{3} + \log y^{4}$
Apply Power Property: Apply the power property to both $\log x^{3}$ and $\log y^{4}$ . Using the power property, we can bring the exponents out in front of the logarithms: $\log x^{3} = 3 \cdot \log x$ $\log y^{4} = 4 \cdot \log y$
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 two terms we found: $\newline$ $3 \cdot \log x + 4 \cdot \log y$

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