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Expand the logarithm fully using the properties of logs. Express the final answer in terms of log x, and log y. log ((x^(3))/(y^(2))) Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log⁡x \log x , and log⁡y \log y .\newlinelog⁡x3y2 \log \frac{x^{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 x , and log⁡y \log y .\newlinelog⁡x3y2 \log \frac{x^{3}}{y^{2}} \newlineAnswer:
  1. Identify Properties: Identify the properties used to expand log⁡(x3y2)\log\left(\frac{x^{3}}{y^{2}}\right). We will use the quotient property of logarithms to separate the numerator and denominator, and the power property to bring the exponents out in front of the logs. Quotient Property: log⁡b(PQ)=log⁡b(P)−log⁡b(Q)\log_b\left(\frac{P}{Q}\right) = \log_b(P) - \log_b(Q) Power Property: log⁡b(Pk)=k⋅log⁡b(P)\log_b(P^k) = k \cdot \log_b(P)
  2. Apply Quotient Property: Apply the quotient property to the logarithm.\newlineUsing the quotient property, we can write log⁡(x3y2)\log\left(\frac{x^{3}}{y^{2}}\right) as log⁡(x3)−log⁡(y2)\log(x^{3}) - \log(y^{2}).
  3. Apply Power Property: Apply the power property to both logarithms.\newlineUsing the power property, we can bring the exponents out in front of each log:\newlinelog⁡(x3)\log(x^{3}) becomes 3×log⁡(x)3 \times \log(x), and log⁡(y2)\log(y^{2}) becomes 2×log⁡(y)2 \times \log(y).
  4. Write Final Form: Write the final expanded form of the logarithm.\newlineCombining the results from steps 22 and 33, we get:\newline3⋅log⁡(x)−2⋅log⁡(y)3 \cdot \log(x) - 2 \cdot \log(y)

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