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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)/(z^(4)root(3)(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⁡xz4y43 \log \frac{x}{z^{4} \sqrt[3]{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⁡xz4y43 \log \frac{x}{z^{4} \sqrt[3]{y^{4}}} \newlineAnswer:
  1. Identify Properties: Identify the properties used to expand the logarithm.\newlineWe will use the quotient property of logarithms to separate the numerator and the denominator, and the power property to bring down the exponents. The properties are as follows:\newlineQuotient Property: log⁡b(PQ)=log⁡bP−log⁡bQ\log_b \left(\frac{P}{Q}\right) = \log_b P - \log_b Q\newlinePower 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 write log⁡(xz4y43)\log\left(\frac{x}{z^{4}\sqrt[3]{y^{4}}}\right) as log⁡(x)−log⁡(z4y43)\log(x) - \log\left(z^{4}\sqrt[3]{y^{4}}\right).
  3. Apply Power Property: Apply the power property to the logarithm of the denominator.\newlineWe have log⁡(z4y43)\log(z^{4}\sqrt[3]{y^{4}}). Since zz is raised to the power of 44, we can bring the exponent down in front of the log, which gives us 4⋅log⁡(z)4 \cdot \log(z). However, we need to be careful with the cube root of y4y^4, which is y43y^{\frac{4}{3}}. We can also bring this exponent down in front of the log, which gives us (43)⋅log⁡(y)\left(\frac{4}{3}\right) \cdot \log(y).
  4. Combine Results: Combine the results from Step 22 and Step 33.\newlineWe now have log⁡(x)−(4×log⁡(z)+(43)×log⁡(y))\log(x) - (4 \times \log(z) + (\frac{4}{3}) \times \log(y)). We need to distribute the negative sign to both terms in the parentheses.
  5. Distribute and Simplify: Distribute the negative sign and simplify.\newlineDistributing the negative sign gives us log⁡(x)−4⋅log⁡(z)−(43)⋅log⁡(y)\log(x) - 4 \cdot \log(z) - \left(\frac{4}{3}\right) \cdot \log(y).

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