Question

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

log
2.5
6.25
logarithm subscript 2.5 baseline 6.25

log
5
0
logarithm subscript 5 baseline 0

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

log
5
1

Answer 1

When exploring the limitations of the values of \( b \) (base) and \( x \) (argument) in logarithmic equations of the form \( \log_b x = L \), we need to ensure that both the base \( b \) and the argument \( x \) meet the necessary conditions for the logarithm to be defined:

1. The base \( b \) must be greater than 0 and not equal to 1; that is, \( b > 0 \) and \( b \neq 1 \).
2. The argument \( x \) must be greater than 0; that is, \( x > 0 \).

With these conditions in mind, let's evaluate the provided logarithmic expressions:

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

2. **\( \log_{5} 0 \)**:
- Base: \( 5 > 0 \) and \( 5 \neq 1 \) (valid)
- Argument: \( 0 \) is not greater than 0 (invalid)
- **This logarithm is undefined.**

3. **\( \log_{\frac{1}{3}} \frac{1}{9} \)**:
- Base: \( \frac{1}{3} > 0 \) and \( \frac{1}{3} \neq 1 \) (valid)
- Argument: \( \frac{1}{9} > 0 \) (valid)
- **This logarithm is defined.**

4. **\( \log_{5} 1 \)**:
- Base: \( 5 > 0 \) and \( 5 \neq 1 \) (valid)
- Argument: \( 1 > 0 \) (valid)
- **This logarithm is defined.**

### Conclusion:
The only logarithm that is undefined is **\( \log_{5} 0 \)** because the argument (0) does not satisfy the requirement of being greater than 0.