The equation you provided in the image appears to be:

$5 \log_{5} 125 + 5 \log_{5} \sqrt{25} + 5 \log_{5} 625$

Let's break this down and solve it step by step:

1. Simplify each logarithmic expression:

• First term: $5 \log_{5} 125$

  • We know that $125 = 5^3$, so $\log_{5} 125 = 3$.
  • This gives us $5 \times 3 = 15$.

• Second term: $5 \log_{5} \sqrt{25}$

  • $\sqrt{25} = 5$, and $\log_{5} 5 = 1$.
  • So, $5 \times 1 = 5$.

• Third term: $5 \log_{5} 625$

  • $625 = 5^4$, so $\log_{5} 625 = 4$.
  • This gives $5 \times 4 = 20$.

2. Add the results:

$15 + 5 + 20 = 40$

Thus, the value of the expression is 40.

Would you like a deeper explanation or clarification on any of these steps? Here are 5 related questions to extend this:

1. What are the properties of logarithms that simplify expressions like this?
2. How do you handle logarithms with bases other than 5?
3. How can logarithms be used to solve exponential equations?
4. Can you generalize this process for any base logarithm?
5. What happens if the terms inside the logarithms are fractions?

Tip: Remember that $\log_b (b^x) = x$ for any base $b$, which is very useful for simplifying logarithmic expressions.