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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^(5))/(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⁡x5y4 \log \frac{x^{5}}{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⁡x5y4 \log \frac{x^{5}}{y^{4}} \newlineAnswer:
  1. Identify Properties: Identify the properties used to expand log⁡(x5y4)\log\left(\frac{x^{5}}{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⁡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⁡(x5y4)\log\left(\frac{x^{5}}{y^{4}}\right) as log⁡(x5)−log⁡(y4)\log(x^{5}) - \log(y^{4}).
  3. Apply Power Property: Apply the power property to both terms.\newlineUsing the power property, we can bring down the exponents in both terms:\newlinelog⁡(x5)\log(x^{5}) becomes 5×log⁡(x)5 \times \log(x), and log⁡(y4)\log(y^{4}) becomes 4×log⁡(y)4 \times \log(y).
  4. Write Final Form: Write the final expanded form.\newlineThe final expanded form of the logarithm is 5⋅log⁡(x)−4⋅log⁡(y)5 \cdot \log(x) - 4 \cdot \log(y).

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