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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 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⁡x5y \log x^{5} y \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⁡x5y \log x^{5} y \newlineAnswer:
  1. Identify Properties: Identify the properties used to expand log⁡x5y\log x^{5}y. We will use the product property of logarithms to separate the terms, and the power property to bring down the exponent. Product property: log⁡b(mn)=log⁡b(m)+log⁡b(n)\log_b(mn) = \log_b(m) + \log_b(n) Power property: log⁡b(mn)=n⋅log⁡b(m)\log_b(m^n) = n \cdot \log_b(m)
  2. Apply Product Property: Apply the product property to log⁡x5y\log x^{5}y. Using the product property, we can write log⁡x5y\log x^{5}y as the sum of two logarithms: log⁡x5+log⁡y\log x^{5} + \log y. log⁡x5y=log⁡x5+log⁡y\log x^{5}y = \log x^{5} + \log y
  3. Apply Power Property: Apply the power property to log⁡x5\log x^{5}. Using the power property, we can bring the exponent outside the logarithm: 5⋅log⁡x5 \cdot \log x. log⁡x5=5⋅log⁡x\log x^{5} = 5 \cdot \log x
  4. Combine Results: Combine the results from Step 22 and Step 33.\newlineWe have already separated log⁡x5y\log x^{5}y into log⁡x5+log⁡y\log x^{5} + \log y and expanded log⁡x5\log x^{5} into 5×log⁡x5 \times \log x. Now we combine them to get the final expanded form.\newlinelog⁡x5y=5×log⁡x+log⁡y\log x^{5}y = 5 \times \log x + \log y

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