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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^(4))) 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⁡x2y4 \log \frac{x^{2}}{y^{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 x , and log⁡y \log y .\newlinelog⁡x2y4 \log \frac{x^{2}}{y^{4}} \newlineAnswer:
  1. Identify Properties: Identify the properties used to expand log⁡(x2y4)\log\left(\frac{x^{2}}{y^{4}}\right). We will use the quotient property of logarithms to separate the numerator and denominator, and the power property to bring down the exponents. Quotient Property: log⁡b(PQ)=log⁡bP−log⁡bQ\log_b \left(\frac{P}{Q}\right) = \log_b P - \log_b Q Power Property: log⁡b(Pk)=k⋅log⁡bP\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⁡(x2y4)\log\left(\frac{x^{2}}{y^{4}}\right) as the difference of two logarithms:\newlinelog⁡(x2y4)=log⁡(x2)−log⁡(y4)\log\left(\frac{x^{2}}{y^{4}}\right) = \log(x^{2}) - \log(y^{4})
  3. Apply Power Property: Apply the power property to both logarithms.\newlineNow we apply the power property to bring down the exponents:\newlinelog⁡(x2)\log(x^{2}) becomes 2⋅log⁡(x)2 \cdot \log(x), and log⁡(y4)\log(y^{4}) becomes 4⋅log⁡(y)4 \cdot \log(y).\newlineSo, log⁡(x2y4)=2⋅log⁡(x)−4⋅log⁡(y)\log\left(\frac{x^{2}}{y^{4}}\right) = 2 \cdot \log(x) - 4 \cdot \log(y)

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