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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 ((zx^(3))/(root(3)(y)))
Answer:

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

\newline

Answer:

Apply Quotient Rule: Apply the quotient rule of logarithms to the expression $\log\left(\frac{zx^{3}}{\sqrt[3]{y}}\right)$ . $\newline$ Quotient rule of logarithm: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ $\newline$ $\log\left(\frac{zx^{3}}{\sqrt[3]{y}}\right) = \log(zx^{3}) - \log(\sqrt[3]{y})$
Apply Product Rule: Apply the product rule of logarithms to the term $\log(zx^{3})$ . $\newline$ Product rule of logarithm: $\log(a \cdot b) = \log(a) + \log(b)$ $\newline$ $\log(zx^{3}) = \log(z) + \log(x^{3})$
Apply Power Rule: Apply the power rule of logarithms to the term $\log(x^{3})$ . $\newline$ Power rule of logarithm: $\log(a^n) = n \cdot \log(a)$ $\newline$ $\log(x^{3}) = 3 \cdot \log(x)$
Apply Power Rule: Apply the power rule of logarithms to the term $\log(\sqrt[3]{y})$ . Since $\sqrt[3]{y}$ is $y$ to the power of $\frac{1}{3}$ , we can write: $\log(\sqrt[3]{y}) = \log(y^{\frac{1}{3}})$ Then apply the power rule: $\log(y^{\frac{1}{3}}) = \frac{1}{3} \cdot \log(y)$
Combine Results: Combine the results from Steps $1$ to $4$ to get the final expanded form. $\newline$ $\log\left(\frac{zx^{3}}{\sqrt[3]{y}}\right) = \log(z) + \log(x^{3}) - \log(\sqrt[3]{y})$ $\newline$ Substitute the results from Steps $3$ and $4$ : $\newline$ $\log\left(\frac{zx^{3}}{\sqrt[3]{y}}\right) = \log(z) + 3 \cdot \log(x) - \left(\frac{1}{3}\right) \cdot \log(y)$

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