The expression in the image is:

$4 \log_5(100) - 8 \log_5(50)$

To simplify this, we will use the properties of logarithms:

1. Change of base rule: $\log_b(x^n) = n \log_b(x)$.
2. Subtraction of logs: $\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)$.

Step-by-step solution:

1. Simplify the individual logarithms:

   $\log_5(100) \quad \text{and} \quad \log_5(50)$

   We can express $100$ and $50$ as powers of 5:

   • $100 = 5^2 \times 4$
   • $50 = 5^1 \times 10$

   Let's approximate these logs first.

   $\log_5(100) \approx 2$ $\log_5(50) \approx 1.43$

2. Apply the coefficients:

   $4 \log_5(100) \approx 4 \times 2 = 8$ $8 \log_5(50) \approx 8 \times 1.43 = 11.44$

3. Subtract the terms:

   $4 \log_5(100) - 8 \log_5(50) \approx 8 - 11.44 = -3.44$

Thus, the approximate value of the expression is:

$-3.44$

Would you like to go over specific details or clarify any step? Here are some related questions:

1. How would the result change if the base of the logarithms was different?
2. Can you simplify logarithmic expressions using different properties of logarithms?
3. What is the significance of the change-of-base formula?
4. How do logarithms with bases other than 10 or e work?
5. What is the graphical interpretation of logarithmic subtraction?

Tip: Always check whether you can factor or simplify the arguments of the logarithms for easier calculation.