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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)root(3)(z))/(x^(2))) 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⁡y5z3x2 \log \frac{y^{5} \sqrt[3]{z}}{x^{2}} \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⁡y5z3x2 \log \frac{y^{5} \sqrt[3]{z}}{x^{2}} \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, the power property to bring the exponents in front of the logs, and the root property to express the cube root as a fractional exponent.
  2. Apply Quotient Property: Apply the quotient property to the logarithm.\newlineThe quotient property of logarithms 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 log⁡(y5z3x2)\log\left(\frac{y^{5}\sqrt[3]{z}}{x^{2}}\right) to get:\newlinelog⁡(y5z3)−log⁡(x2)\log(y^{5}\sqrt[3]{z}) - \log(x^{2})
  3. Apply Power Property: Apply the power property to the logarithms.\newlineThe power property of logarithms states that log⁡b(Pk)=klog⁡b(P)\log_b(P^k) = k\log_b(P). We apply this to both terms:\newline5log⁡(y)+log⁡(z13)−2log⁡(x)5\log(y) + \log(z^{\frac{1}{3}}) - 2\log(x)
  4. Apply Root Property: Apply the root property to the logarithm of zz. The cube root of zz can be written as z(1/3)z^{(1/3)}, so we have already applied the root property when we wrote log⁡(z(1/3))\log(z^{(1/3)}) in the previous step.
  5. Combine Results: Combine the results to get the final expanded form.\newlineThe final expanded form of the logarithm is:\newline5log⁡(y)+(13)log⁡(z)−2log⁡(x)5\log(y) + \left(\frac{1}{3}\right)\log(z) - 2\log(x)

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