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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 ((root(3)(z^(4)))/(y^(4)x^(5)))
Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x, \log y$ , and $\log z$ . $\newline$ $\log \frac{\sqrt[3]{z^{4}}}{y^{4} x^{5}}$ $\newline$ 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{\sqrt[3]{z^{4}}}{y^{4} x^{5}}

\newline

Answer:

Apply Quotient Rule: Apply the quotient rule of logarithms to the expression $\log\left(\frac{\sqrt[3]{z^{4}}}{y^{4}x^{5}}\right)$ . Quotient rule of logarithm: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ $\log\left(\frac{\sqrt[3]{z^{4}}}{y^{4}x^{5}}\right) = \log(\sqrt[3]{z^{4}}) - \log(y^{4}x^{5})$
Apply Product Rule: Apply the product rule of logarithms to the expression $\log(y^{4}x^{5})$ . $\newline$ Product rule of logarithm: $\log(a \cdot b) = \log(a) + \log(b)$ $\newline$ $\log(y^{4}x^{5}) = \log(y^{4}) + \log(x^{5})$
Apply Power Rule: Apply the power rule of logarithms to the expressions $\log(\sqrt[3]{z^{4}})$ , $\log(y^{4})$ , and $\log(x^{5})$ . Power rule of logarithm: $\log(a^{n}) = n \cdot \log(a)$ $\log(\sqrt[3]{z^{4}}) = \log(z^{\frac{4}{3}}) = \frac{4}{3} \cdot \log(z)$ $\log(y^{4}) = 4 \cdot \log(y)$ $\log(x^{5}) = 5 \cdot \log(x)$
Substitute Results: Substitute the results from Step $3$ into the expression from Step $1$ . $\newline$ $\log(\sqrt[3]{z^{4}}) - \log(y^{4}x^{5}) = \frac{4}{3} \cdot \log(z) - (4 \cdot \log(y) + 5 \cdot \log(x))$
Distribute Negative Sign: Distribute the negative sign to the terms inside the parenthesis. $\newline$ $(\frac{4}{3}) \cdot \log(z) - (4 \cdot \log(y) + 5 \cdot \log(x)) = (\frac{4}{3}) \cdot \log(z) - 4 \cdot \log(y) - 5 \cdot \log(x)$

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