To convert \( \log_{25} 200 \) to an equivalent expression using the change of base formula, we use the formula:
\[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]
For \( \log_{25} 200 \), we can choose any base \( c \), but a common choice is base 10 (logarithm) or the natural logarithm (ln).
Applying the change of base formula gives:
\[ \log_{25} 200 = \frac{\log 200}{\log 25} \]
or
\[ \log_{25} 200 = \frac{\ln 200}{\ln 25} \]
Looking at the provided responses:
- \( \frac{\log 200}{\log 25} \) - This is correct.
- \( \frac{\log 25}{\log 200} \) - This is incorrect (the order is reversed).
- \( \frac{\log 200}{\log_5 25} \) - This isn't necessarily equivalent without further calculation, as it uses a different base.
- \( \log_{200} \ln 25 \) - This is incorrect.
Therefore, the correct equivalent expression is:
\[ \frac{\log 200}{\log 25} \]