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

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

\newline

Answer:

Apply Quotient Rule: Apply the quotient rule of logarithms to the expression $\log\left(\frac{x^{4}\sqrt{y^{3}}}{z^{5}}\right)$ . The quotient rule of logarithms states that $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ .
Apply Quotient Rule: Apply the quotient rule to the given logarithm. $\log\left(\frac{x^{4}\sqrt{y^{3}}}{z^{5}}\right) = \log(x^{4}\sqrt{y^{3}}) - \log(z^{5})$
Apply Product Rule: Apply the product rule of logarithms to $\log(x^{4}\sqrt{y^{3}})$ . The product rule of logarithms states that $\log(a*b) = \log(a) + \log(b)$ . Since $\sqrt{y^{3}}$ is the same as $y^{\frac{3}{2}}$ , we can write: $\log(x^{4}\sqrt{y^{3}}) = \log(x^{4}) + \log(y^{\frac{3}{2}})$
Apply Power Rule: Apply the power rule of logarithms to each term. $\newline$ The power rule of logarithms states that $\log(a^n) = n\log(a)$ . $\newline$ $\log(x^{4}) = 4\log(x)$ $\newline$ $\log(y^{\frac{3}{2}}) = \left(\frac{3}{2}\right)\log(y)$ $\newline$ $\log(z^{5}) = 5\log(z)$
Substitute Results: Substitute the results from the power rule back into the expression from Step $2$ . $\newline$ $\log\left(\frac{x^{4}\sqrt{y^{3}}}{z^{5}}\right) = (4\log(x)) + \left(\frac{3}{2}\right)\log(y) - (5\log(z))$

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