Question

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


logarithm subscript 2.5 baseline 6.25


logarithm subscript 5 baseline 1


logarithm subscript 5 baseline 0


log¹/³ 1/9

Answer 1

In logarithmic equations of the form \( \log_b x = L \), there are specific conditions that must be met for the logarithm to be defined:

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

With these criteria in mind, let's evaluate each provided logarithm to determine which one is undefined:

1. **\( \log_{2.5} 6.25 \)**: Here, the base \( b = 2.5 \) is positive and not equal to 1, and \( x = 6.25 \) is positive. Thus, this logarithm is defined.

2. **\( \log_5 1 \)**: The base \( b = 5 \) is positive and not equal to 1, and \( x = 1 \) is positive. Thus, this logarithm is defined.

3. **\( \log_5 0 \)**: The base \( b = 5 \) is positive and not equal to 1, but \( x = 0 \) is not positive. Therefore, this logarithm is undefined.

4. **\( \log_{\frac{1}{3}} \frac{1}{9} \)**: The base \( b = \frac{1}{3} \) is positive but equal to \( \frac{1}{3} \neq 1\), and \( x = \frac{1}{9} \) is positive. Thus, this logarithm is defined.

Based on this analysis, the logarithm that is undefined is:

**\( \log_5 0 \)**