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

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

log51
logarithm subscript 5 baseline 1

log2.56.25
logarithm subscript 2.5 baseline 6.25

log50

Answer 1

To determine which logarithm is undefined among the given options, we can analyze the values of \( b \) (the base) and \( x \) (the argument of the logarithm) for each logarithmic expression according to the rules for logarithms:

1. The base \( b \) must be greater than 0 and cannot be equal to 1 (i.e., \( b > 0 \) and \( b \neq 1 \)).
2. The argument \( x \) must be greater than 0 (i.e., \( x > 0 \)).

Let’s evaluate each option:

1. **\( \log_{13} 19 \)**: Here, \( b = 13 \) (valid) and \( x = 19 \) (valid). **This expression is defined.**

2. **\( \log_{\frac{1}{3}} \frac{1}{9} \)**: The base \( b = \frac{1}{3} \) (valid since \( b > 0 \) and \( b \neq 1 \)), and the argument \( x = \frac{1}{9} \) (valid since \( x > 0 \)). **This expression is defined.**

3. **\( \log_{5} 1 \)**: The base \( b = 5 \) (valid), and the argument \( x = 1 \) (valid since \( x > 0 \)). **This expression is defined.**

4. **\( \log_{2.5} 6.25 \)**: The base \( b = 2.5 \) (valid), and the argument \( x = 6.25 \) (valid since \( x > 0 \)). **This expression is defined.**

5. **\( \log_{5} 0 \)**: The base \( b = 5 \) (valid), but the argument \( x = 0 \) (invalid, as \( x > 0 \) is required). **This expression is undefined.**

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

**\( \log_{5} 0 \)**.