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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^(2))/(y))
Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of logx \log x , and logy \log y .\newlinelogx2y \log \frac{x^{2}}{y} \newlineAnswer:

Full solution

Q. Expand the logarithm fully using the properties of logs. Express the final answer in terms of logx \log x , and logy \log y .\newlinelogx2y \log \frac{x^{2}}{y} \newlineAnswer:
  1. Identify Properties: Identify the properties used to expand log(x2y)\log\left(\frac{x^{2}}{y}\right). We have a logarithm of a quotient and a power. We will use the quotient property and the power property of logarithms to expand this expression. Quotient Property: logb(PQ)=logbPlogbQ\log_b \left(\frac{P}{Q}\right) = \log_b P - \log_b Q Power Property: logb(Pk)=klogbP\log_b (P^k) = k \cdot \log_b P
  2. Apply Quotient Property: Apply the quotient property to the logarithm log(x2y)\log\left(\frac{x^{2}}{y}\right). Using the quotient property, we can write log(x2y)\log\left(\frac{x^{2}}{y}\right) as log(x2)log(y)\log(x^{2}) - \log(y).
  3. Apply Power Property: Apply the power property to the logarithm log(x2)\log(x^{2}). Using the power property, we can write log(x2)\log(x^{2}) as 2×log(x)2 \times \log(x).
  4. Combine Results: Combine the results from Step 22 and Step 33 to get the final expanded form.\newlineThe final expanded form is 2log(x)log(y)2 \cdot \log(x) - \log(y).

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