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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)))/(z^(3)y^(2))) 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⁡x5z3y2 \log \frac{\sqrt{x^{5}}}{z^{3} y^{2}} \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⁡x5z3y2 \log \frac{\sqrt{x^{5}}}{z^{3} y^{2}} \newlineAnswer:
  1. Apply Quotient Rule: We start by applying the quotient rule of logarithms, which states that log⁡(ab)=log⁡(a)−log⁡(b)\log(\frac{a}{b}) = \log(a) - \log(b). In this case, we have a=x5a = \sqrt{x^5} and b=z3y2b = z^3y^2.
  2. Apply Power Rule: Next, we apply the power rule of logarithms, which states that log⁡(ab)=b⋅log⁡(a)\log(a^b) = b\cdot\log(a). For the numerator, we have x5\sqrt{x^5} which is x5/2x^{5/2}, and for the denominator, we have z3z^3 and y2y^2.
  3. Rewrite Using Product Rule: We can now rewrite the logarithm as log⁡(x52)−(log⁡(z3)+log⁡(y2))\log(x^{\frac{5}{2}}) - (\log(z^3) + \log(y^2)) by applying the product rule of logarithms, which states that log⁡(ab)=log⁡(a)+log⁡(b)\log(ab) = \log(a) + \log(b), to the denominator.
  4. Apply Power Rule: We apply the power rule to each term: log⁡(x52)\log(x^{\frac{5}{2}}) becomes 52⋅log⁡(x)\frac{5}{2}\cdot\log(x), log⁡(z3)\log(z^3) becomes 3⋅log⁡(z)3\cdot\log(z), and log⁡(y2)\log(y^2) becomes 2⋅log⁡(y)2\cdot\log(y).
  5. Subtract Logs: Subtracting the logs in the denominator from the log in the numerator, we get (52)log⁡(x)−3log⁡(z)−2log⁡(y)(\frac{5}{2})\log(x) - 3\log(z) - 2\log(y).
  6. Final Expanded Form: This is the fully expanded form of the original logarithm in terms of log⁡x\log x, log⁡y\log y, and log⁡z\log z.

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