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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)/(z^(5)x)) 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⁡yz5x \log \frac{y}{z^{5} x} \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⁡yz5x \log \frac{y}{z^{5} x} \newlineAnswer:
  1. Identify Properties: Identify the properties used to expand log⁡(yz5x)\log\left(\frac{y}{z^{5}x}\right). We will use the quotient property and the power property of logarithms to expand the given expression. Quotient Property: log⁡b(PQ)=log⁡bP−log⁡bQ\log_{b} \left(\frac{P}{Q}\right) = \log_{b} P - \log_{b} Q Power 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 separate the numerator and the denominator of the fraction inside the logarithm.\newlinelog⁡(yz5x)=log⁡(y)−log⁡(z5x)\log\left(\frac{y}{z^{5}x}\right) = \log(y) - \log(z^{5}x)
  3. Separate z5z^{5} and xx: Apply the quotient property again to separate z5z^{5} and xx. Now we have log⁡(z5x)\log(z^{5}x), which is a product inside the logarithm. We can use the quotient property again to separate them. log⁡(z5x)=log⁡(z5)+log⁡(x)\log(z^{5}x) = \log(z^{5}) + \log(x)
  4. Apply Power Property: Apply the power property to the term log⁡(z5)\log(z^{5}). Using the power property, we can take the exponent out in front of the log. log⁡(z5)=5×log⁡(z)\log(z^{5}) = 5 \times \log(z)
  5. Substitute Expanded Term: Substitute the expanded term back into the original expression.\newlineNow we can replace log⁡(z5x)\log(z^{5}x) in our original expression with the expanded terms.\newlinelog⁡(yz5x)=log⁡(y)−(5⋅log⁡(z)+log⁡(x))\log\left(\frac{y}{z^{5}x}\right) = \log(y) - (5 \cdot \log(z) + \log(x))
  6. Distribute Negative Sign: Distribute the negative sign to both terms in the parentheses.\newlineWhen we distribute the negative sign, we get:\newlinelog⁡(yz5x)=log⁡(y)−5×log⁡(z)−log⁡(x)\log\left(\frac{y}{z^{5}x}\right) = \log(y) - 5 \times \log(z) - \log(x)

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