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.
A. log
2.5
​
6.25
B. log
3
​

9
1
​

C. log
5
​
0
D. log
5
​
1

Answer 1

To explore the limitations of the values of \( b \) and \( x \) in equations of the form \( \log_b x = L \), we need to consider the following rules for logarithms:

1. The base \( b \) must be positive and cannot be equal to 1 (\( b > 0 \) and \( b \neq 1 \)).
2. The argument \( x \) must be positive (\( x > 0 \)).

Let's evaluate each option based on these rules:

### A. \( \log_{2.5} 6.25 \)
- The base \( b = 2.5 \) is positive and not equal to 1.
- The argument \( x = 6.25 \) is positive.
- Therefore, this logarithm is defined.

### B. \( \log_3 \left( \frac{9}{1} \right) \)
- The base \( b = 3 \) is positive and not equal to 1.
- The argument \( x = \frac{9}{1} = 9 \) is positive.
- Therefore, this logarithm is defined.

### C. \( \log_5 0 \)
- The base \( b = 5 \) is positive and not equal to 1.
- The argument \( x = 0 \) is not positive (it must be greater than 0).
- Therefore, this logarithm is undefined.

### D. \( \log_5 1 \)
- The base \( b = 5 \) is positive and not equal to 1.
- The argument \( x = 1 \) is positive.
- Therefore, this logarithm is defined.

### Summary of Results:
- **A:** Defined
- **B:** Defined
- **C:** **Undefined** (since \( x = 0 \))
- **D:** Defined

**Conclusion:** The logarithm that is undefined is **C. \( \log_5 0 \)**.