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Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log x.

log 4x^(3)
Answer:

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

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Product property of logarithms

Full solution

Q. Expand the logarithm fully using the properties of logs. Express the final answer in terms of

\log x

.

\newline

\log 4 x^{3}

\newline

Answer:

Identify Properties: Identify the properties of logarithms that can be used to expand $\log 4x^{3}$ . The expression $4x^{3}$ is a product of $4$ and $x^{3}$ . We can use the product property of logarithms to separate these two parts. Additionally, we can use the power property of logarithms to bring down the exponent on $x$ .
Apply Product Property: Apply the product property to separate the constant from the variable. $\newline$ The product property of logarithms states that $\log(ab) = \log(a) + \log(b)$ . We can apply this to $\log(4x^{3})$ to get: $\newline$ $\log(4x^{3}) = \log(4) + \log(x^{3})$
Apply Power Property: Apply the power property to the logarithm of $x^{3}$ . The power property of logarithms states that $\log(a^{b}) = b\log(a)$ . We can apply this to $\log(x^{3})$ to get: $\log(x^{3}) = 3\log(x)$
Substitute Expanded Log: Substitute the expanded $\log(x^{3})$ back into the equation from Step $2$ . $\log(4x^{3}) = \log(4) + 3\log(x)$
Recognize Constant Log: Recognize that $\log(4)$ is a constant and cannot be simplified further. Since $\log(4)$ is the logarithm of a constant, it remains as is.

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