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

1 answer

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. \( \log_5 0 \).