To expand the expression $\log\left(\frac{x^5}{y \cdot z^5}\right)$, we can use the laws of logarithms. Here’s the step-by-step process:

Step 1: Apply the Quotient Rule

The Quotient Rule states that $\log\left(\frac{A}{B}\right) = \log(A) - \log(B)$.

So, $\log\left(\frac{x^5}{y \cdot z^5}\right) = \log(x^5) - \log(y \cdot z^5)$

Step 2: Apply the Product Rule

The Product Rule states that $\log(A \cdot B) = \log(A) + \log(B)$.

Thus, $\log(x^5) - \log(y \cdot z^5) = \log(x^5) - (\log(y) + \log(z^5))$

Simplifying further, $= \log(x^5) - \log(y) - \log(z^5)$

Step 3: Apply the Power Rule

The Power Rule states that $\log(A^k) = k \cdot \log(A)$.

Applying this to each term with an exponent: $= 5 \cdot \log(x) - \log(y) - 5 \cdot \log(z)$

Final Answer

The expanded form of the expression is: $5 \cdot \log(x) - \log(y) - 5 \cdot \log(z)$

Would you like further clarification on any steps or details?

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Here are some related questions:

1. How would you expand $\log\left(\frac{a^3 b}{c^2 d^4}\right)$?
2. Can you simplify $\log(x^2 y^3 z^4)$ using the laws of logarithms?
3. How would you approach expanding $\ln\left(\frac{m^7}{n \cdot p^2}\right)$ with natural logarithms?
4. What are the differences between using logarithms with base 10 and base $e$?
5. How does changing the base of a logarithm affect the expansion process?

Tip: Remember that the laws of logarithms—Product, Quotient, and Power—are essential tools for simplifying complex logarithmic expressions.