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$\frac{1}{2}\log(x)+2\log(3)$

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Quotient property of logarithms

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Q.

\frac{1}{2}\log(x)+2\log(3)

Identify Properties: Identify the properties used to expand the logarithms. $\newline$ The first term $(\frac{1}{2})\log(x)$ suggests the use of the power property of logarithms, which states that $\log_b(a^n) = n\log_b(a)$ . The second term $2\log(3)$ also suggests the use of the power property.
Apply Power Property $1/2$ : Apply the power property to the first term $1/2$ $\log(x)$ . According to the power property, we can move the coefficient inside the logarithm as an exponent of the argument. Therefore, $1/2$ $\log(x)$ becomes \(\newline\log(x^{1/2})\).
Apply Power Property ( $2$ ): Apply the power property to the second term $2\log(3)$ . Similarly, we can move the coefficient inside the logarithm as an exponent of the argument. Therefore, $2\log(3)$ becomes $\log(3^2)$ .
Simplify Expressions: Simplify the expressions inside the logarithms. $\newline$ $x^{\frac{1}{2}}$ is the square root of $x$ , and $3^2$ is $9$ . So, $\log(x^{\frac{1}{2}})$ becomes $\log(\sqrt{x})$ , and $\log(3^2)$ becomes $\log(9)$ .
Write Final Form: Write the final expanded form of the logarithm. $\newline$ The expanded form of the logarithm is $\log(\sqrt{x}) + \log(9)$ .

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