To solve the expression \( \log_5(125x^3) \), we'll break it down step by step using logarithmic properties.

### Step-by-Step Solution

1. **Express 125 as a Power of 5:**
   The number 125 can be expressed as \( 5^3 \). So:
   \[
   125x^3 = (5^3) \times x^3
   \]

2. **Rewrite the Logarithm:**
   Substitute back into the logarithmic expression:
   \[
   \log_5(125x^3) = \log_5((5^3) \times x^3)
   \]

3. **Apply the Product Rule for Logarithms:**
   Recall that \(\log_a(bc) = \log_a(b) + \log_a(c)\):
   \[
   \log_5((5^3) \times x^3) = \log_5(5^3) + \log_5(x^3)
   \]

4. **Simplify Each Logarithm:**
   - For \(\log_5(5^3)\), apply the power rule \(\log_a(a^b) = b\):
     \[
     \log_5(5^3) = 3
     \]
   - For \(\log_5(x^3)\), also apply the power rule \(\log_a(b^c) = c \cdot \log_a(b)\):
     \[
     \log_5(x^3) = 3 \log_5(x)
     \]

5. **Combine the Results:**
   \[
   \log_5(125x^3) = 3 + 3\log_5(x)
   \]

### Final Answer:
\[
\log_5(125x^3) = 3 + 3\log_5(x)
\]

Would you like more details on any part of this solution, or do you have any questions?

### Related Questions to Expand Understanding:
1. How would you solve \(\log_5(25x^2)\)?
2. What is the result of \(\log_5(625)\)?
3. How do you apply the power rule to simplify \(\log_5(x^4)\)?
4. Can you explain the properties of logarithms, such as the product rule and power rule, in more depth?
5. How would you solve \(\log_5(5x^5) + \log_5(x^3)\)?

### Tip:
Always try to express numbers in the logarithm as powers of the base if possible—this helps simplify the calculation significantly.

Learn how to solve log5(125x^3) step by step. This problem involves using the product and power rules of logarithms, breaking down 125 into powers of 5, and simplifying the expression for clarity. Perfect for students in Grades 10-12.

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