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 $e$).