Explore the limitations of the values of b and x in equations of the form logbx=L

Question

by determining which logarithm is undefined.(1 point)
Responses

log50
logarithm subscript 5 baseline 0

log51
logarithm subscript 5 baseline 1

log2.56.25
logarithm subscript 2.5 baseline 6.25

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

Answer 1

In logarithmic equations of the form \( \log_b(x) = L \), there are several key restrictions on the values of the base \( b \) and the argument \( x \):

1. The base \( b \) must be a positive number, and \( b \neq 1 \).
2. The argument \( x \) must be a positive number.

Let's evaluate each of the given logarithms with respect to these rules:

1. log₅(0): This logarithm is undefined because the argument (0) is not positive. Therefore, \( \log_5(0) \) is undefined.

2. log₅(1): This logarithm is defined because \( b \) (5) is positive and \( x \) (1) is positive. In fact, \( \log_5(1) = 0 \) since any base to the power of 0 equals 1.

3. log₂.₅(6.25): This logarithm is also defined because \( b \) (2.5) is positive (and not equal to 1) and \( x \) (6.25) is positive.

4. log₁/₃(1/9): This logarithm is defined too because \( b \) (1/3) is positive and not equal to 1, and \( x \) (1/9) is positive.

In summary, the only logarithm in the list that is undefined is:

• log₅(0), because \( 0 \) is not a positive number.