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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 ((y^(5)x)/(root(3)(z^(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⁡y5xz43 \log \frac{y^{5} x}{\sqrt[3]{z^{4}}} \newlineAnswer:

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Quotient property of logarithmsright chevron icon

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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⁡y5xz43 \log \frac{y^{5} x}{\sqrt[3]{z^{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 denominator, the power property to bring down exponents, and the property for the logarithm of a product to separate the terms in the numerator.
  2. Apply Quotient Property: Apply the quotient property to the logarithm.\newlineThe quotient property states that log⁡b(PQ)=log⁡b(P)−log⁡b(Q)\log_b\left(\frac{P}{Q}\right) = \log_b(P) - \log_b(Q). We apply this to our logarithm:\newlinelog⁡(y5xz43)=log⁡(y5x)−log⁡(z43)\log\left(\frac{y^{5}x}{\sqrt[3]{z^{4}}}\right) = \log(y^{5}x) - \log(\sqrt[3]{z^{4}})
  3. Apply Product Property: Apply the product property to the logarithm of the numerator.\newlineThe product property states that log⁡b(PQ)=log⁡b(P)+log⁡b(Q)\log_b(PQ) = \log_b(P) + \log_b(Q). We apply this to the logarithm of the numerator:\newlinelog⁡(y5x)=log⁡(y5)+log⁡(x)\log(y^{5}x) = \log(y^{5}) + \log(x)
  4. Apply Power Property y5y^5: Apply the power property to the logarithm of y5y^5. The power property states that log⁡b(Pk)=k⋅log⁡b(P)\log_b(P^k) = k \cdot \log_b(P). We apply this to log⁡(y5)\log(y^{5}): \(\newline\log(y^{5}) = 5 \cdot \log(y)\)
  5. Apply Power Property z4z^4: Apply the power property to the logarithm of the denominator.\newlineWe need to express the cube root of z4z^4 as z43z^{\frac{4}{3}} and then apply the power property:\newlinelog⁡(z43)=log⁡(z43)=43⋅log⁡(z)\log(\sqrt[3]{z^{4}}) = \log(z^{\frac{4}{3}}) = \frac{4}{3} \cdot \log(z)
  6. Combine Properties: Combine all the properties to write the final expanded form.\newlineWe combine the results from steps 22 to 55 to get the final expanded form:\newlinelog⁡(y5xz43)=(log⁡(y5)+log⁡(x))−log⁡(z43)\log\left(\frac{y^{5}x}{\sqrt[3]{z^{4}}}\right) = (\log(y^{5}) + \log(x)) - \log(\sqrt[3]{z^{4}})\newline = (5⋅log⁡(y)+log⁡(x))−(43)⋅log⁡(z)(5 \cdot \log(y) + \log(x)) - \left(\frac{4}{3}\right) \cdot \log(z)\newline = 5⋅log⁡(y)+log⁡(x)−(43)⋅log⁡(z)5 \cdot \log(y) + \log(x) - \left(\frac{4}{3}\right) \cdot \log(z)

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