Question

Explore the limitations of the values of b and x in equations of the form logbx=L
by determining which logarithm is undefined.(1 point)
Responses

log51
logarithm subscript 5 baseline 1

log50
logarithm subscript 5 baseline 0

log2.56.25
logarithm subscript 2.5 baseline 6.25

log1319
logarithm subscript Start Fraction1 over 3 End Fraction baseline Start Fraction 1 over 9 End Fraction

Answer 1

To analyze the values of \(b\) and \(x\) in logarithmic equations of the form \(\log_b(x) = L\) and determine which logarithm is undefined, we need to recall the properties of logarithms. Specifically, for a logarithm \(\log_b(x)\) to be defined, two conditions must be met:

1. **Base condition**: The base \(b\) must be positive and not equal to 1 (i.e., \(b > 0\) and \(b \neq 1\)).
2. **Argument condition**: The argument \(x\) must be positive (i.e., \(x > 0\)).

Now, let's evaluate each of the provided logarithmic statements to identify which one is undefined:

1. **\(\log_5(1)\)**:
- Base \(b = 5\) (valid, as \(b > 0\) and \(b \neq 1\)).
- Argument \(x = 1\) (valid, as \(x > 0\)).
- This logarithm is defined.

2. **\(\log_5(0)\)**:
- Base \(b = 5\) (valid, as \(b > 0\) and \(b \neq 1\)).
- Argument \(x = 0\) (invalid, as \(x\) must be greater than 0).
- This logarithm is undefined.

3. **\(\log_{2.5}(6.25)\)**:
- Base \(b = 2.5\) (valid, as \(b > 0\) and \(b \neq 1\)).
- Argument \(x = 6.25\) (valid, as \(x > 0\)).
- This logarithm is defined.

4. **\(\log_{1/3}(1/9)\)**:
- Base \(b = \frac{1}{3}\) (valid, as it is greater than 0 but less than 1).
- Argument \(x = \frac{1}{9}\) (valid, as \(x > 0\)).
- This logarithm is defined.

Based on the evaluations, the only logarithm that is **undefined** among the options presented is:

- **\(\log_5(0)\)**.