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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 ((y^(3))/(root(3)(z^(5))x^(2)))
Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x, \log y$ , and $\log z$ . $\newline$ $\log \frac{y^{3}}{\sqrt[3]{z^{5}} x^{2}}$ $\newline$ Answer:

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Evaluate definite integrals using the power rule

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{y^{3}}{\sqrt[3]{z^{5}} x^{2}}

\newline

Answer:

Use Quotient Rule: Use the quotient rule for logarithms, which states that $\log(\frac{a}{b}) = \log(a) - \log(b)$ , to separate the numerator and the denominator. $\newline$ \log\left(\frac{y^{\(3\)}}{\sqrt[\(3\)]{z^{\(5\)}}x^{\(2\)}}\right) = \log(y^{\(3\)}) - \log(\sqrt[\(3]{z^{ $5$ }}x^{ $2$ })
Apply Power Rule: Apply the power rule for logarithms, which states that $\log(a^n) = n\log(a)$ , to the term $\log(y^{3})$ . $\newline$ $\log(y^{3}) = 3\log(y)$
Convert Denominator to Powers: The denominator contains a cube root and a square, which can be written as powers: $\sqrt[3]{z^{5}} = z^{\frac{5}{3}}$ and $x^{2}$ . $\log(\sqrt[3]{z^{5}}x^{2}) = \log(z^{\frac{5}{3}}x^{2})$
Use Product Rule: Use the product rule for logarithms, which states that $\log(ab) = \log(a) + \log(b)$ , to separate the terms in the denominator. $\newline$ $\log(z^{\frac{5}{3}}x^{2}) = \log(z^{\frac{5}{3}}) + \log(x^{2})$
Apply Power Rule: Apply the power rule for logarithms to both terms in the denominator. $\newline$ $\log(z^{\frac{5}{3}}) = \frac{5}{3}\log(z)$ $\newline$ $\log(x^{2}) = 2\log(x)$
Combine Results: Combine the results using the properties of logarithms. $\newline$ $\log\left(\frac{y^{3}}{\sqrt[3]{z^{5}}x^{2}}\right) = 3\log(y) - \left(\frac{5}{3}\log(z) + 2\log(x)\right)$
Distribute Negative Sign: Distribute the negative sign to both terms in the parentheses. $\newline$ $3\log(y) - \left(\frac{5}{3}\right)\log(z) - 2\log(x)$

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