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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^(3)))/(z^(5)y)) 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⁡x3z5y \log \frac{\sqrt{x^{3}}}{z^{5} y} \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⁡x3z5y \log \frac{\sqrt{x^{3}}}{z^{5} y} \newlineAnswer:
  1. Apply Quotient Rule: We are given the logarithm log⁡(x3z5y)\log\left(\frac{\sqrt{x^{3}}}{z^{5}y}\right). To expand this logarithm fully, we will use the properties of logarithms, which include the quotient rule, the power rule, and the square root as a power of 12\frac{1}{2}.
  2. Apply Product Rule: First, apply the quotient rule log⁡(ab)=log⁡(a)−log⁡(b)\log(\frac{a}{b}) = \log(a) - \log(b) to the given logarithm.\newlinelog⁡(x3z5y)=log⁡(x3)−log⁡(z5y)\log\left(\frac{\sqrt{x^{3}}}{z^{5}y}\right) = \log(\sqrt{x^{3}}) - \log(z^{5}y)
  3. Apply Power Rule to Exponents: Next, apply the product rule log⁡(ab)=log⁡(a)+log⁡(b)\log(ab) = \log(a) + \log(b) to the second term of the expression.log⁡(x3)−log⁡(z5y)=log⁡(x3)−(log⁡(z5)+log⁡(y))\log(\sqrt{x^{3}}) - \log(z^{5}y) = \log(\sqrt{x^{3}}) - (\log(z^{5}) + \log(y))
  4. Apply Power Rule to Term: Now, apply the power rule log⁡(an)=nlog⁡(a)\log(a^{n}) = n\log(a) to the terms with exponents.log⁡(x3)−(log⁡(z5)+log⁡(y))=(12)log⁡(x3)−(5log⁡(z)+log⁡(y))\log(\sqrt{x^{3}}) - (\log(z^{5}) + \log(y)) = (\frac{1}{2})\log(x^{3}) - (5\log(z) + \log(y))
  5. Simplify the Expression: Finally, apply the power rule again to the term (12)⋅log⁡(x3)(\frac{1}{2})\cdot\log(x^{3}).(\frac{\(1\)}{\(2\)})\cdot\log(x^{\(3\)}) - (\(5\cdot\log(z) + \log(y)) = (\frac{11}{22})\cdot(33\cdot\log(x)) - (55\cdot\log(z) + \log(y))
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