QuestionJuly 15, 2025

Write expression log((x^8y^7)/(z^20)) as a sum or difference of logarithms with no exponents. Simplify your answer completely. log((x^8y^7)/(z^20))=square

Write expression log((x^8y^7)/(z^20)) as a sum or difference of logarithms with no exponents. Simplify your answer completely. log((x^8y^7)/(z^20))=square
Write expression log((x^8y^7)/(z^20)) as a sum or difference of logarithms with no exponents. Simplify your answer completely. log((x^8y^7)/(z^20))=square

Solution
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Answer

8 \cdot \log(x) + 7 \cdot \log(y) - 20 \cdot \log(z) Explanation 1. Apply the Quotient Rule Use **\log(\frac{a}{b}) = \log(a) - \log(b)** to separate the numerator and denominator. \log\left(\frac{x^8 y^7}{z^{20}}\right) = \log(x^8 y^7) - \log(z^{20}) 2. Apply the Product Rule Use **\log(ab) = \log(a) + \log(b)** to separate the terms in the numerator. \log(x^8 y^7) = \log(x^8) + \log(y^7) 3. Apply the Power Rule Use **\log(a^b) = b \cdot \log(a)** to remove exponents. \log(x^8) = 8 \cdot \log(x) \log(y^7) = 7 \cdot \log(y) \log(z^{20}) = 20 \cdot \log(z) 4. Combine All Parts Substitute back into the expression from Step 1. 8 \cdot \log(x) + 7 \cdot \log(y) - 20 \cdot \log(z)

Explanation

1. Apply the Quotient Rule<br /> Use **$\log(\frac{a}{b}) = \log(a) - \log(b)$** to separate the numerator and denominator.<br /> $\log\left(\frac{x^8 y^7}{z^{20}}\right) = \log(x^8 y^7) - \log(z^{20})$<br /><br />2. Apply the Product Rule<br /> Use **$\log(ab) = \log(a) + \log(b)$** to separate the terms in the numerator.<br /> $\log(x^8 y^7) = \log(x^8) + \log(y^7)$<br /><br />3. Apply the Power Rule<br /> Use **$\log(a^b) = b \cdot \log(a)$** to remove exponents.<br /> $\log(x^8) = 8 \cdot \log(x)$<br /> $\log(y^7) = 7 \cdot \log(y)$<br /> $\log(z^{20}) = 20 \cdot \log(z)$<br /><br />4. Combine All Parts<br /> Substitute back into the expression from Step 1.<br /> $8 \cdot \log(x) + 7 \cdot \log(y) - 20 \cdot \log(z)$
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