Question

Which of the following logarithmic expressions is undefined?

A
39\log_39log
3
​
9

B
1.32.8561\log_{1.3}2.8561log
1.3
​
2.8561

C
3.19.61\log_{3.1}9.61log
3.1
​
9.61

D
2−5\log_2-5log
2
​
−5

Answer 1

To determine which logarithmic expression is undefined, we must analyze the base and the argument of each logarithm. A logarithm is defined only when its base is positive and not equal to 1, and its argument is positive.

Let's evaluate each option:

**A. \( \log_{3} 9 \)**
- Base: 3 (which is positive and not equal to 1).
- Argument: 9 (which is positive).
- This logarithm is defined.

**B. \( \log_{1.3} 2.8561 \)**
- Base: 1.3 (which is positive and not equal to 1).
- Argument: 2.8561 (which is positive).
- This logarithm is defined.

**C. \( \log_{3.1} 9.61 \)**
- Base: 3.1 (which is positive and not equal to 1).
- Argument: 9.61 (which is positive).
- This logarithm is defined.

**D. \( \log_{2} -5 \)**
- Base: 2 (which is positive and not equal to 1).
- Argument: -5 (which is negative).
- Logarithms with negative arguments are undefined.

Thus, the logarithmic expression that is undefined is **D. \( \log_{2} -5 \)**.