Questions: Use the properties of logarithms to expand log((y^5 * sqrt[3](x))/z^4). Each logarithm in your answer should involve only one variable. Assume that all variables are positive.

Use the properties of logarithms to expand log((y^5 * sqrt[3](x))/z^4). Each logarithm in your answer should involve only one variable. Assume that all variables are positive.
Transcript text: Use the properties of logarithms to expand $\log \left(\frac{y^{5} \sqrt[3]{x}}{z^{4}}\right)$. Each logarithm in your answer should Involve only one variable. Assume that all variables are positive.
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Solution

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Solution Steps

Step 1: Apply the Quotient Rule

The quotient rule for logarithms states that \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\). Applying this rule to the given expression:

\[ \log \left(\frac{y^{5} \sqrt[3]{x}}{z^{4}}\right) = \log(y^{5} \sqrt[3]{x}) - \log(z^{4}) \]

Step 2: Apply the Product Rule

The product rule for logarithms states that \(\log(ab) = \log(a) + \log(b)\). Applying this rule to \(\log(y^{5} \sqrt[3]{x})\):

\[ \log(y^{5} \sqrt[3]{x}) = \log(y^{5}) + \log(\sqrt[3]{x}) \]

Step 3: Apply the Power Rule

The power rule for logarithms states that \(\log(a^b) = b \log(a)\). Applying this rule to each term:

  • For \(\log(y^{5})\):

\[ \log(y^{5}) = 5 \log(y) \]

  • For \(\log(\sqrt[3]{x})\), note that \(\sqrt[3]{x} = x^{1/3}\):

\[ \log(\sqrt[3]{x}) = \log(x^{1/3}) = \frac{1}{3} \log(x) \]

  • For \(\log(z^{4})\):

\[ \log(z^{4}) = 4 \log(z) \]

Step 4: Combine All Terms

Substitute the expanded terms back into the expression:

\[ \log \left(\frac{y^{5} \sqrt[3]{x}}{z^{4}}\right) = 5 \log(y) + \frac{1}{3} \log(x) - 4 \log(z) \]

Final Answer

\[ \boxed{5 \log(y) + \frac{1}{3} \log(x) - 4 \log(z)} \]

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