The problem requires rewriting the logarithmic expression $\log_2 \left( \frac{x^8 y^{1/3}}{z^7} \right)$ as a sum and/or difference of logarithms, expressing the powers as factors.

Here are the steps:

1. Apply the logarithmic property for division: $\log_b \left( \frac{A}{B} \right) = \log_b(A) - \log_b(B)$ This gives: $\log_2 \left( \frac{x^8 y^{1/3}}{z^7} \right) = \log_2 \left( x^8 y^{1/3} \right) - \log_2 \left( z^7 \right)$

2. Apply the logarithmic property for multiplication: $\log_b(A \cdot B) = \log_b(A) + \log_b(B)$ This gives: $\log_2 \left( x^8 y^{1/3} \right) = \log_2 \left( x^8 \right) + \log_2 \left( y^{1/3} \right)$

   Substituting this back, we have: $\log_2 \left( \frac{x^8 y^{1/3}}{z^7} \right) = \log_2 \left( x^8 \right) + \log_2 \left( y^{1/3} \right) - \log_2 \left( z^7 \right)$

3. Apply the logarithmic property for exponents: $\log_b(A^n) = n \cdot \log_b(A)$ This gives: $\log_2 \left( x^8 \right) = 8 \cdot \log_2(x)$ $\log_2 \left( y^{1/3} \right) = \frac{1}{3} \cdot \log_2(y)$ $\log_2 \left( z^7 \right) = 7 \cdot \log_2(z)$

   Substituting these into the equation, we get: $\log_2 \left( \frac{x^8 y^{1/3}}{z^7} \right) = 8 \cdot \log_2(x) + \frac{1}{3} \cdot \log_2(y) - 7 \cdot \log_2(z)$

Final Answer:

$\log_2 \left( \frac{x^8 y^{1/3}}{z^7} \right) = 8 \cdot \log_2(x) + \frac{1}{3} \cdot \log_2(y) - 7 \cdot \log_2(z)$

Would you like a step-by-step breakdown of the properties used or clarification on any part of this solution?

Related Questions:

1. What are the key properties of logarithms?
2. How do you simplify expressions involving logarithms of products and quotients?
3. What are the common mistakes made while simplifying logarithmic expressions?
4. How is this process related to exponential equations?
5. Can logarithmic expressions be solved when the base is not explicitly given?

Tip:

Always write down the logarithmic properties you’re using at each step to avoid errors in simplification!