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

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Quotient property of logarithms

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

\log x, \log y

, and

\log z

\newline

\log \frac{x^{2}}{\sqrt{z} y^{2}}

\newline

Answer:

Apply Quotient Rule: Apply the quotient rule of logarithms to the expression $\log\left(\frac{x^{2}}{\sqrt{y}\cdot y^{2}}\right)$ . Quotient rule of logarithms: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ $\log\left(\frac{x^{2}}{\sqrt{y}\cdot y^{2}}\right) = \log(x^{2}) - \log(\sqrt{y}\cdot y^{2})$
Apply Product Rule: Apply the product rule of logarithms to the denominator part of the expression $\log(\sqrt{y}\cdot y^{2})$ . $\newline$ Product rule of logarithms: $\log(a\cdot b) = \log(a) + \log(b)$ $\newline$ $\log(\sqrt{y}\cdot y^{2}) = \log(\sqrt{y}) + \log(y^{2})$
Apply Power Rule: Apply the power rule of logarithms to the expressions $\log(x^{2})$ , $\log(\sqrt{y})$ , and $\log(y^{2})$ .
Power rule of logarithms: $\log(a^{n}) = n\log(a)$
$\log(x^{2}) = 2\log(x)$
$\log(\sqrt{y}) = \log(y^{\frac{1}{2}}) = \frac{1}{2}\log(y)$
$\log(y^{2}) = 2\log(y)$
Substitute Results: Substitute the results from Step $3$ into the expression from Step $1$ . $\newline$ $\log\left(\frac{x^{2}}{\sqrt{y}y^{2}}\right) = 2\log(x) - \left(\frac{1}{2}\log(y) + 2\log(y)\right)$
Distribute and Combine: Distribute the negative sign and combine the logarithmic terms involving $\log(y)$ . $2\log(x) - (\frac{1}{2})\log(y) - 2\log(y) = 2\log(x) - (\frac{1}{2} + 2)\log(y)$
Simplify Expression: Simplify the expression by combining the coefficients for $\log(y)$ . $2\log(x) - \left(\frac{1}{2} + 2\right)\log(y) = 2\log(x) - \left(\frac{5}{2}\right)\log(y)$
Write Final Form: Write the final expanded form of the logarithm. $\newline$ The final expanded form is $2\log(x) - \left(\frac{5}{2}\right)\log(y)$ .