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

log ((xz^(2))/(y))
Answer:

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

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Quotient property of logarithms

Full solution

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 z^{2}}{y}

\newline

Answer:

Identify Properties: Identify the properties used to expand $\log\left(\frac{xz^{2}}{y}\right)$ . We will use the quotient property and the product property of logarithms to expand the given expression. Quotient Property: $\log_b \left(\frac{P}{Q}\right) = \log_b P - \log_b Q$ Product Property: $\log_b (PQ) = \log_b P + \log_b Q$ Power Property: $\log_b (P^k) = k \cdot \log_b P$
Apply Quotient Property: Apply the quotient property to the given logarithm. $\newline$ Using the quotient property, we can separate the numerator and the denominator of the fraction inside the logarithm. $\newline$ $\log\left(\frac{xz^{2}}{y}\right) = \log(xz^{2}) - \log(y)$
Apply Product Property: Apply the product property to the logarithm of the numerator. $\newline$ Now we will separate the $x$ and $z^{2}$ terms using the product property. $\newline$ $log(xz^{2}) = log(x) + log(z^{2})$
Apply Power Property: Apply the power property to the logarithm of $z^{2}$ . We will now apply the power property to the $\log(z^{2})$ term to bring down the exponent. $\log(z^{2}) = 2 \cdot \log(z)$
Combine Results: Combine the results from Steps $2$ , $3$ , and $4$ to get the final expanded form. $\newline$ Substitute the expanded form of $\log(z^{2})$ from Step $4$ into the equation from Step $3$ , and then combine it with the result from Step $2$ . $\newline$ $\log\left(\frac{xz^{2}}{y}\right) = (\log(x) + 2 \cdot \log(z)) - \log(y)$
Simplify Expression: Simplify the expression to get the final answer. $\log\left(\frac{xz^{2}}{y}\right) = \log(x) + 2 \cdot \log(z) - \log(y)$

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