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

Expand the logarithm fully using the properties of logs. Express the final answer in terms of log⁡x,log⁡y \log x, \log y , and log⁡z \log z .\newlinelog⁡x2y3z \log \frac{x^{2} \sqrt[3]{y}}{z} \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⁡y \log x, \log y , and log⁡z \log z .\newlinelog⁡x2y3z \log \frac{x^{2} \sqrt[3]{y}}{z} \newlineAnswer:
  1. Identify Properties: Identify the properties of logarithms that will be used to expand the given logarithm.\newlineThe properties of logarithms that we will use are the quotient property, the power property, and the property for the logarithm of a root.
  2. Apply Quotient Property: Apply the quotient property to the given logarithm.\newlineThe quotient property states that log⁡b(PQ)=log⁡b(P)−log⁡b(Q)\log_b(\frac{P}{Q}) = \log_b(P) - \log_b(Q). Therefore, we can write log⁡(x2y3z)\log(\frac{x^{2}\sqrt[3]{y}}{z}) as log⁡(x2y3)−log⁡(z)\log(x^{2}\sqrt[3]{y}) - \log(z).
  3. Apply Power Property: Apply the power property to the term log⁡(x2y3)\log(x^{2}\sqrt[3]{y}). The power property states that log⁡b(Pk)=klog⁡b(P)\log_b(P^k) = k\log_b(P). We can apply this to the term x2x^2 to get 2log⁡(x)+log⁡(y3)2\log(x) + \log(\sqrt[3]{y}).
  4. Apply Root Property: Apply the property for the logarithm of a root to the term log⁡(y3)\log(\sqrt[3]{y}). The property for the logarithm of a root states that log⁡b(Pk)=1klog⁡b(P)\log_b(\sqrt[k]{P}) = \frac{1}{k}\log_b(P). We can apply this to the term y3\sqrt[3]{y} to get 13log⁡(y)\frac{1}{3}\log(y).
  5. Combine Results: Combine the results from the previous steps to get the final expanded form.\newlineWe have 2log⁡(x)+13log⁡(y)−log⁡(z)2\log(x) + \frac{1}{3}\log(y) - \log(z) as the expanded form of the given logarithm.

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