Which of the following is true?

Options

A. `log_(11)1650gtlog_(13)1950`
B. `log_(11)1650ltlog_(13)1950`
C. `log_(11)1650=log_(13)1950`
D. None of these

Answer

To determine which of the options is true, we need to evaluate the logarithmic expressions given in the question: \( \log_{1650} 11 \) and \( \log_{1950} 13 \).

### Step-by-Step Solution:

1. **Understanding Logarithm Basics**:
The logarithm \( \log_b a \) answers the question: "To what power must the base \( b \) be raised to obtain \( a \)?"

2. **Finding \( \log_{1650} 11 \)**:
We need to find the value of \( \log_{1650} 11 \).
- We can estimate this by finding powers of 11:
- \( 11^2 = 121 \)
- \( 11^3 = 1331 \)
- \( 11^4 = 14641 \)
- Since \( 1650 \) is between \( 1331 \) and \( 14641 \), we can conclude that \( 2 < \log_{1650} 11 < 4 \).

3. **Finding \( \log_{1950} 13 \)**:
Similarly, we need to find \( \log_{1950} 13 \).
- We can estimate this by finding powers of 13:
- \( 13^2 = 169 \)
- \( 13^3 = 2197 \)
- Since \( 1950 \) is between \( 169 \) and \( 2197 \), we can conclude that \( 2 < \log_{1950} 13 < 3 \).

4. **Comparing the Values**:
- From our estimates, we have:
- \( \log_{1650} 11 \) is approximately \( 3 \) (between \( 2 \) and \( 4 \)).
- \( \log_{1950} 13 \) is approximately \( 2 \) (between \( 2 \) and \( 3 \)).
- Therefore, we can conclude that:
\[ \log_{1650} 11 > \log_{1950} 13 \]

5. **Conclusion**:
The first option states that \( \log_{1650} 11 > \log_{1950} 13 \), which we have confirmed to be true.

### Final Answer:
The correct option is that \( \log_{1650} 11 \) is greater than \( \log_{1950} 13 \).