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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 ((sqrt(x^(5)))/(y^(3)z^(4))) 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⁡x5y3z4 \log \frac{\sqrt{x^{5}}}{y^{3} z^{4}} \newlineAnswer:

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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⁡x5y3z4 \log \frac{\sqrt{x^{5}}}{y^{3} z^{4}} \newlineAnswer:
  1. Apply Quotient Rule: Apply the quotient rule for logarithms, which states that log⁡(ab)=log⁡(a)−log⁡(b)\log\left(\frac{a}{b}\right) = \log(a) - \log(b).log⁡(x5y3z4)=log⁡(x5)−log⁡(y3z4)\log\left(\frac{\sqrt{x^{5}}}{y^{3}z^{4}}\right) = \log(\sqrt{x^{5}}) - \log(y^{3}z^{4})
  2. Apply Product Rule: Apply the product rule for logarithms, which states that log⁡(ab)=log⁡(a)+log⁡(b)\log(ab) = \log(a) + \log(b), to the denominator of the original expression.\newlinelog⁡(y3z4)=log⁡(y3)+log⁡(z4)\log(y^{3}z^{4}) = \log(y^{3}) + \log(z^{4})
  3. Apply Power Rule: Apply the power rule for logarithms, which states that log⁡(an)=nlog⁡(a)\log(a^n) = n\log(a), to each logarithm.\newlinelog⁡(x5)=12⋅log⁡(x5)=52⋅log⁡(x)\log(\sqrt{x^{5}}) = \frac{1}{2} \cdot \log(x^{5}) = \frac{5}{2} \cdot \log(x)\newlinelog⁡(y3)=3⋅log⁡(y)\log(y^{3}) = 3 \cdot \log(y)\newlinelog⁡(z4)=4⋅log⁡(z)\log(z^{4}) = 4 \cdot \log(z)
  4. Substitute Results: Substitute the results from Step 33 back into the expression from Step 11. \log\left(\frac{\sqrt{x^{\(5\)}}}{y^{\(3\)}z^{\(4\)}}\right) = \left(\frac{\(5\)}{\(2\)}\right) \cdot \log(x) - \left(\(3 \cdot \log(y) + 44 \cdot \log(z)\right)
  5. Distribute Negative Sign: Distribute the negative sign to both terms in the parentheses.\newlinelog⁡(x5y3z4)=52⋅log⁡(x)−3⋅log⁡(y)−4⋅log⁡(z)\log\left(\frac{\sqrt{x^{5}}}{y^{3}z^{4}}\right) = \frac{5}{2} \cdot \log(x) - 3 \cdot \log(y) - 4 \cdot \log(z)

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