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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))/(y^(2)z^(3))) 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⁡x5y2z3 \log \frac{x^{5}}{y^{2} z^{3}} \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⁡x5y2z3 \log \frac{x^{5}}{y^{2} z^{3}} \newlineAnswer:
  1. Identify Properties: Identify the properties used to expand log⁡(x5y2z3)\log\left(\frac{x^{5}}{y^{2}z^{3}}\right). We will use the quotient property of logarithms to separate the numerator and denominator, and the power property to bring down the exponents. Quotient Property: log⁡b(PQ)=log⁡b(P)−log⁡b(Q)\log_b\left(\frac{P}{Q}\right) = \log_b(P) - \log_b(Q) Power Property: log⁡b(Pk)=k⋅log⁡b(P)\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⁡(x5y2z3)\log\left(\frac{x^{5}}{y^{2}z^{3}}\right) as log⁡(x5)−log⁡(y2z3)\log(x^{5}) - \log(y^{2}z^{3}).
  3. Apply Power Property: Apply the power property to the terms with exponents.\newlineUsing the power property, we can write log⁡(x5)\log(x^{5}) as 5×log⁡(x)5 \times \log(x) and log⁡(y2z3)\log(y^{2}z^{3}) as log⁡(y2)+log⁡(z3)\log(y^{2}) + \log(z^{3}) because of the product inside the logarithm.
  4. Expand Product: Expand the logarithm of the product y2z3y^{2}z^{3}. Using the product property of logarithms, which states that log⁡b(PQ)=log⁡b(P)+log⁡b(Q)\log_b(PQ) = \log_b(P) + \log_b(Q), we can write log⁡(y2z3)\log(y^{2}z^{3}) as log⁡(y2)+log⁡(z3)\log(y^{2}) + \log(z^{3}).
  5. Apply Power Property: Apply the power property to the remaining terms with exponents.\newlineUsing the power property again, we can write log⁡(y2)\log(y^{2}) as 2×log⁡(y)2 \times \log(y) and log⁡(z3)\log(z^{3}) as 3×log⁡(z)3 \times \log(z).
  6. Combine Terms: Combine all the terms to get the final expanded form.\newlineNow we combine all the terms from the previous steps to get the final expanded form: 5×log⁡(x)−(2×log⁡(y)+3×log⁡(z))5 \times \log(x) - (2 \times \log(y) + 3 \times \log(z)).
  7. Distribute Negative Sign: Distribute the negative sign to the terms in the parentheses.\newlineDistributing the negative sign gives us 5⋅log⁡(x)−2⋅log⁡(y)−3⋅log⁡(z)5 \cdot \log(x) - 2 \cdot \log(y) - 3 \cdot \log(z).

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