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Expand the logarithm. Assume all expressions exist and are well-defined. $\newline$ Write your answer as a sum or difference of common logarithms or multiples of common logarithms. The inside of each logarithm must be a distinct constant or variable. $\newline$ $\log\left(\frac{z}{xy}\right)$ $\newline$ ______

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Quotient property of logarithms

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Q. Expand the logarithm. Assume all expressions exist and are well-defined.

\newline

Write your answer as a sum or difference of common logarithms or multiples of common logarithms. The inside of each logarithm must be a distinct constant or variable.

\newline

\log\left(\frac{z}{xy}\right)

\newline

______

Apply quotient rule of logarithms: Apply the quotient rule of logarithms to the expression $\log\left(\frac{z}{xy}\right)$ . $\newline$ The quotient rule of logarithms states that $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ . Therefore, we can rewrite $\log\left(\frac{z}{xy}\right)$ as $\log(z) - \log(xy)$ .
Apply product rule of logarithms: Apply the product rule of logarithms to the expression $\log(xy)$ . $\newline$ The product rule of logarithms states that $\log(a \cdot b) = \log(a) + \log(b)$ . Therefore, we can rewrite $\log(xy)$ as $\log(x) + \log(y)$ .
Combine results from Step $1$ and Step $2$ : Combine the results from Step $1$ and Step $2$ . $\newline$ We have $\log(z)$ from Step $1$ and $\log(x) + \log(y)$ from Step $2$ . Combining these, we get $\log(z) - (\log(x) + \log(y))$ . Distributing the negative sign, we get $\log(z) - \log(x) - \log(y)$ .

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