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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 ((zx^(2))/(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⁡zx2y4 \log \frac{z 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⁡y \log x, \log y , and log⁡z \log z .\newlinelog⁡zx2y4 \log \frac{z x^{2}}{y^{4}} \newlineAnswer:
  1. Apply Quotient Rule: Apply the quotient rule of logarithms to the expression log⁡(zx2y4)\log\left(\frac{zx^{2}}{y^{4}}\right). Quotient rule of logarithm: log⁡(ab)=log⁡(a)−log⁡(b)\log\left(\frac{a}{b}\right) = \log(a) - \log(b) log⁡(zx2y4)=log⁡(zx2)−log⁡(y4)\log\left(\frac{zx^{2}}{y^{4}}\right) = \log(zx^{2}) - \log(y^{4})
  2. Apply Product Rule: Apply the product rule of logarithms to the term log⁡(zx2)\log(zx^{2}).\newlineProduct rule of logarithm: log⁡(a⋅b)=log⁡(a)+log⁡(b)\log(a \cdot b) = \log(a) + \log(b)\newlinelog⁡(zx2)=log⁡(z)+log⁡(x2)\log(zx^{2}) = \log(z) + \log(x^{2})
  3. Apply Power Rule: Apply the power rule of logarithms to the terms log⁡(x2)\log(x^{2}) and log⁡(y4)\log(y^{4}).\newlinePower rule of logarithm: log⁡(an)=n⋅log⁡(a)\log(a^{n}) = n \cdot \log(a)\newlinelog⁡(x2)=2⋅log⁡(x)\log(x^{2}) = 2 \cdot \log(x)\newlinelog⁡(y4)=4⋅log⁡(y)\log(y^{4}) = 4 \cdot \log(y)
  4. Substitute Results: Substitute the results from Step 33 back into the equation from Step 11.\newlinelog⁡(zx2y4)=log⁡(z)+2⋅log⁡(x)−4⋅log⁡(y)\log\left(\frac{z x^{2}}{y^{4}}\right) = \log(z) + 2 \cdot \log(x) - 4 \cdot \log(y)
  5. Distribute and Simplify: Distribute the negative sign to 4⋅log⁡(y)4 \cdot \log(y) and simplify the expression.log⁡(zx2y4)=log⁡(z)+2⋅log⁡(x)−4⋅log⁡(y)\log\left(\frac{zx^{2}}{y^{4}}\right) = \log(z) + 2 \cdot \log(x) - 4 \cdot \log(y)

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