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To rewrite \( \log_{20}{125} \) using the change-of-base formula, let's go through the options and see which ones match the change-of-base form.

### Change-of-Base Formula
The change-of-base formula states that:
\[
\log_b{a} = \frac{\log_c{a}}{\log_c{b}}
\]
where \( c \) can be any positive base (commonly 10 or \( e \)).

So, applying this formula to \( \log_{20}{125} \), we get:
\[
\log_{20}{125} = \frac{\log{125}}{\log{20}} \quad \text{or} \quad \frac{\ln{125}}{\ln{20}}
\]

### Checking the Options
Let's evaluate each option:

1. **Option 1:** \( \log{\frac{125}{20}} \)  
   - This does not match the change-of-base formula. Instead, it represents the logarithm of the quotient \( \frac{125}{20} \), which is not equivalent to \( \log_{20}{125} \).
   
2. **Option 2:** \( \frac{\log{125}}{\log{20}} \)  
   - This matches the change-of-base formula with base 10. It is a correct representation of \( \log_{20}{125} \).
   
3. **Option 3:** \( \frac{\ln{125}}{\ln{20}} \)  
   - This also matches the change-of-base formula but uses the natural logarithm \( \ln \) instead of \( \log \). This is also a correct representation of \( \log_{20}{125} \).
   
4. **Option 4:** \( \log_{125}{(20)} \)  
   - This does not match the change-of-base formula. Instead, it represents a logarithm with base 125, which is different from \( \log_{20}{125} \).

### Conclusion
The correct answers are:
- \( \frac{\log{125}}{\log{20}} \)
- \( \frac{\ln{125}}{\ln{20}} \)

These are the options that apply the change-of-base formula correctly.

Would you like further clarification on any of these steps? Here are some additional questions related to this topic:

1. How do we apply the change-of-base formula to logarithms with bases other than 10 or \( e \)?
2. Why is the change-of-base formula useful when calculating logarithms with unusual bases?
3. What is the difference between \( \log \) and \( \ln \) in terms of base?
4. Can the change-of-base formula be used with bases other than \( 10 \) or \( e \), like base 2?
5. How would you verify that two logarithmic expressions are equivalent?

**Tip**: Use the change-of-base formula to simplify calculations when a calculator only supports base 10 or natural logarithms (base \( e \)).

Use the Change-of-Base formula and write the expression that is equivalent to log_20(125). Choose all that apply. Do not solve!

Understand how to use the Change-of-Base formula to rewrite log_20(125) in different forms. This problem explores logarithmic concepts and the Change-of-Base formula, suitable for high school students (Grades 9-12).

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