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

log2.56.25
logarithm subscript 2.5 baseline 6.25

log50
logarithm subscript 5 baseline 0

log1319

Answer 1

To explore the limitations of the values of \( b \) (the base of the logarithm) and \( x \) (the argument of the logarithm) in equations of the form \( \log_b(x) = L \), we need to consider the basic properties of logarithms:

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

Now, let's analyze the provided logarithm expressions:

1. **\( \log_5(1) \)**: This is defined because the base \( 5 \) is positive and not equal to 1, and \( 1 > 0 \). So, this value is defined.

2. **\( \log_{2.5}(6.25) \)**: This is also defined because the base \( 2.5 \) is positive and not equal to 1, and \( 6.25 > 0 \). So, this value is defined.

3. **\( \log_5(0) \)**: This is **undefined** because the argument \( 0 \) is not greater than 0. Logarithms cannot take non-positive values as arguments, so this logarithm is undefined.

4. **\( \log_{13}(19) \)**: This is defined because the base \( 13 \) is positive and not equal to 1, and \( 19 > 0 \). So, this value is defined.

In conclusion, the logarithm that is undefined in your list is:

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