To change \( \log_{25} 200 \) to an equivalent expression, we can use the change of base formula for logarithms. The change of base formula states:
\[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]
where \( c \) is any positive number (often we use 10 or \( e \)).
Applying this to \( \log_{25} 200 \):
\[ \log_{25} 200 = \frac{\log_{10} 200}{\log_{10} 25} \quad \text{(using base 10)} \]
or we can use the natural logarithm \( \ln \):
\[ \log_{25} 200 = \frac{\ln 200}{\ln 25} \]
Now let's identify which of the choices provided matches this equivalent expression:
- Option A: \(\frac{\log 200}{\log 25}\) (This is indeed correct as it represents the change of base formula).
- Option B: \(\frac{\log 200}{\log_5 25}\) (This is incorrect; it uses a different base for the denominator).
- Option C: \(\frac{\log 200}{\ln 25}\) (This is partially correct if using a different base, but it does not match the original expression exactly).
- Option D: \(\frac{\log 25}{\log 200}\) (This is incorrect; it's inverted).
The correct choice to change \( \log_{25} 200 \) to an equivalent expression is:
Option A: \(\frac{\log 200}{\log 25}\)