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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^(5)z)/(y^(4))) 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⁡x5zy4 \log \frac{x^{5} z}{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⁡y \log x, \log y , and log⁡z \log z .\newlinelog⁡x5zy4 \log \frac{x^{5} z}{y^{4}} \newlineAnswer:
  1. Identify Properties: Identify the properties used to expand log⁡(x5zy4)\log\left(\frac{x^{5}z}{y^{4}}\right). We will use the quotient property and the power property of logarithms to expand the given expression. 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 given logarithm.\newlineUsing the quotient property, we can separate the numerator and the denominator:\newlinelog⁡(x5zy4)=log⁡(x5z)−log⁡(y4)\log\left(\frac{x^{5}z}{y^{4}}\right) = \log(x^{5}z) - \log(y^{4})
  3. Apply Product Property: Apply the product property to the logarithm of the numerator.\newlineThe product property states that log⁡b(P×Q)=log⁡bP+log⁡bQ\log_b (P \times Q) = \log_b P + \log_b Q.\newlinelog⁡(x5z)=log⁡(x5)+log⁡(z)\log(x^{5}z) = \log(x^{5}) + \log(z)
  4. Apply Power Property: Apply the power property to the logarithms with exponents.\newlineUsing the power property, we can bring the exponents out in front of the logarithms:\newlinelog⁡(x5)=5⋅log⁡(x)\log(x^{5}) = 5 \cdot \log(x)\newlinelog⁡(y4)=4⋅log⁡(y)\log(y^{4}) = 4 \cdot \log(y)
  5. Substitute Expanded Forms: Substitute the expanded forms back into the original expression.\newlineNow we substitute the expanded forms from steps 33 and 44 back into the expression from step 22:\newlinelog⁡(x5zy4)=(5⋅log⁡(x))+log⁡(z)−(4⋅log⁡(y))\log\left(\frac{x^{5}z}{y^{4}}\right) = (5 \cdot \log(x)) + \log(z) - (4 \cdot \log(y))

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