The expression you're working with is:

$\log_2 \left( \frac{4x}{5yz} \right)$

To rewrite this as a sum or difference of logarithms, you can apply the logarithmic properties:

1. Quotient Rule: $\log_b \left( \frac{a}{c} \right) = \log_b(a) - \log_b(c)$
2. Product Rule: $\log_b(ab) = \log_b(a) + \log_b(b)$
3. Power Rule: $\log_b(a^n) = n \log_b(a)$

Let's break it down step by step:

• Apply the Quotient Rule to the fraction $\frac{4x}{5yz}$:

  $\log_2 \left( \frac{4x}{5yz} \right) = \log_2(4x) - \log_2(5yz)$

• Now, apply the Product Rule to the terms $4x$ and $5yz$.

  For $\log_2(4x)$: $\log_2(4x) = \log_2(4) + \log_2(x)$

  For $\log_2(5yz)$: $\log_2(5yz) = \log_2(5) + \log_2(y) + \log_2(z)$

• Now put everything together: $\log_2 \left( \frac{4x}{5yz} \right) = \left( \log_2(4) + \log_2(x) \right) - \left( \log_2(5) + \log_2(y) + \log_2(z) \right)$

• Finally, simplify the constants: $\log_2(4) = 2 \quad \text{(since } 4 = 2^2\text{)}$ So the final expression is: $\log_2 \left( \frac{4x}{5yz} \right) = 2 + \log_2(x) - \log_2(5) - \log_2(y) - \log_2(z)$

That's the logarithmic expression rewritten as a sum or difference of logarithms.

Would you like to discuss the steps further or need any clarifications?

Here are some related questions you might find interesting:

1. What is the meaning and application of the power rule for logarithms?
2. How do logarithmic functions relate to exponential functions?
3. What happens if the logarithm has a base of 10, also known as the common logarithm?
4. How can you solve logarithmic equations algebraically?
5. How can logarithms be applied in real-world scenarios, such as in measuring sound intensity (decibels)?

Tip: Remember to simplify logarithmic expressions by using the quotient, product, and power rules whenever you encounter fractions, products, or exponents in logarithms.