Question

Which of the following correctly changes log25 200
to an equivalent expression?(1 point)
Responses

log200/log25
Start Fraction logarithm 200 over logarithm 25 End Fraction

log200/log525
Start Fraction logarithm 200 over logarithm subscript 5 baseline 25 End Fraction

log200/ln25
Start Fraction logarithm 200 over l n 25 End Fraction

log25/log200

Answer 1

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:** \(\frac{\log 200}{\log 25}\)