Let's break down the expression \(\log(\sqrt{125} \cdot p) \times \log_p(25)\).

### Step 1: Simplify the Expression

1. **Simplifying \(\log(\sqrt{125} \cdot p)\)**:
   - \(\sqrt{125}\) can be simplified as \( \sqrt{125} = \sqrt{5^3} = 5^{3/2} \).
   - So, \(\sqrt{125} \cdot p = 5^{3/2} \cdot p\).

   Now, the logarithm of the product is:

   \[
   \log(5^{3/2} \cdot p) = \log(5^{3/2}) + \log(p)
   \]

   Since \(\log(5^{3/2}) = \frac{3}{2} \log(5)\), we have:

   \[
   \log(5^{3/2} \cdot p) = \frac{3}{2} \log(5) + \log(p)
   \]

2. **Simplifying \(\log_p(25)\)**:
   - \(25\) can be written as \(5^2\).
   - Using the change of base formula, \(\log_p(25)\) can be rewritten as:

   \[
   \log_p(25) = \frac{\log(25)}{\log(p)}
   \]

   Since \(25 = 5^2\), \(\log(25) = 2 \log(5)\):

   \[
   \log_p(25) = \frac{2 \log(5)}{\log(p)}
   \]

### Step 2: Combine the Expressions

Now, multiply the two simplified expressions together:

\[
\left(\frac{3}{2} \log(5) + \log(p)\right) \times \frac{2 \log(5)}{\log(p)}
\]

Expanding the multiplication:

\[
= \frac{3}{2} \cdot \frac{2 \log(5) \cdot \log(5)}{\log(p)} + \frac{2 \log(5) \cdot \log(p)}{\log(p)}
\]

Simplifying each term:

- The first term simplifies to:

  \[
  \frac{3}{2} \cdot 2 \cdot \frac{\log^2(5)}{\log(p)} = 3 \cdot \frac{\log^2(5)}{\log(p)}
  \]

- The second term simplifies to:

  \[
  2 \log(5)
  \]

So the final expression is:

\[
3 \cdot \frac{\log^2(5)}{\log(p)} + 2 \log(5)
\]

This is the simplified form of the original expression.

Would you like more details or have any questions?

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Here are five related questions:
1. How does the change of base formula work for logarithms?
2. Can logarithms of different bases be added directly?
3. What are the properties of logarithms when dealing with products and powers?
4. How would the expression change if the base \(p\) were a specific number like 10 or \(e\)?
5. How do logarithms behave when taking the logarithm of a square root?

**Tip:** Remember that when simplifying logarithmic expressions, always apply the logarithm laws step by step to avoid mistakes.

Learn how to simplify the logarithmic expression log(sqrt(125) * p) × log_p(25) step-by-step. This problem involves logarithms, including the change of base formula and properties of logarithms. Suitable for high school students studying advanced algebra.

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