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

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log⁡x,log⁡y \log x, \log y , and log⁡z \log z .\newlinelog⁡z43y4x5 \log \frac{\sqrt[3]{z^{4}}}{y^{4} x^{5}} \newlineAnswer:

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Quotient property of logarithmsright chevron icon

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Q. Expand the logarithm fully using the properties of logs. Express the final answer in terms of log⁡x,log⁡y \log x, \log y , and log⁡z \log z .\newlinelog⁡z43y4x5 \log \frac{\sqrt[3]{z^{4}}}{y^{4} x^{5}} \newlineAnswer:
  1. Apply Quotient Rule: Apply the quotient rule of logarithms to the expression log⁡(z43y4x5)\log\left(\frac{\sqrt[3]{z^{4}}}{y^{4}x^{5}}\right). Quotient rule of logarithm: log⁡(ab)=log⁡(a)−log⁡(b)\log\left(\frac{a}{b}\right) = \log(a) - \log(b) log⁡(z43y4x5)=log⁡(z43)−log⁡(y4x5)\log\left(\frac{\sqrt[3]{z^{4}}}{y^{4}x^{5}}\right) = \log(\sqrt[3]{z^{4}}) - \log(y^{4}x^{5})
  2. Apply Product Rule: Apply the product rule of logarithms to the expression log⁡(y4x5)\log(y^{4}x^{5}).\newlineProduct rule of logarithm: log⁡(a⋅b)=log⁡(a)+log⁡(b)\log(a \cdot b) = \log(a) + \log(b)\newlinelog⁡(y4x5)=log⁡(y4)+log⁡(x5)\log(y^{4}x^{5}) = \log(y^{4}) + \log(x^{5})
  3. Apply Power Rule: Apply the power rule of logarithms to the expressions log⁡(z43)\log(\sqrt[3]{z^{4}}), log⁡(y4)\log(y^{4}), and log⁡(x5)\log(x^{5}). Power rule of logarithm: log⁡(an)=n⋅log⁡(a)\log(a^{n}) = n \cdot \log(a) log⁡(z43)=log⁡(z43)=43⋅log⁡(z)\log(\sqrt[3]{z^{4}}) = \log(z^{\frac{4}{3}}) = \frac{4}{3} \cdot \log(z) log⁡(y4)=4⋅log⁡(y)\log(y^{4}) = 4 \cdot \log(y) log⁡(x5)=5⋅log⁡(x)\log(x^{5}) = 5 \cdot \log(x)
  4. Substitute Results: Substitute the results from Step 33 into the expression from Step 11.\newlinelog⁡(z43)−log⁡(y4x5)=43⋅log⁡(z)−(4⋅log⁡(y)+5⋅log⁡(x))\log(\sqrt[3]{z^{4}}) - \log(y^{4}x^{5}) = \frac{4}{3} \cdot \log(z) - (4 \cdot \log(y) + 5 \cdot \log(x))
  5. Distribute Negative Sign: Distribute the negative sign to the terms inside the parenthesis.\newline(43)⋅log⁡(z)−(4⋅log⁡(y)+5⋅log⁡(x))=(43)⋅log⁡(z)−4⋅log⁡(y)−5⋅log⁡(x)(\frac{4}{3}) \cdot \log(z) - (4 \cdot \log(y) + 5 \cdot \log(x)) = (\frac{4}{3}) \cdot \log(z) - 4 \cdot \log(y) - 5 \cdot \log(x)

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