To analyze the values of \(b\) and \(x\) in logarithmic equations of the form \(\log_b(x) = L\) and determine which logarithm is undefined, we need to recall the properties of logarithms. Specifically, for a logarithm \(\log_b(x)\) to be defined, two conditions must be met:
- Base condition: The base \(b\) must be positive and not equal to 1 (i.e., \(b > 0\) and \(b \neq 1\)).
- Argument condition: The argument \(x\) must be positive (i.e., \(x > 0\)).
Now, let's evaluate each of the provided logarithmic statements to identify which one is undefined:
-
\(\log_5(1)\):
- Base \(b = 5\) (valid, as \(b > 0\) and \(b \neq 1\)).
- Argument \(x = 1\) (valid, as \(x > 0\)).
- This logarithm is defined.
-
\(\log_5(0)\):
- Base \(b = 5\) (valid, as \(b > 0\) and \(b \neq 1\)).
- Argument \(x = 0\) (invalid, as \(x\) must be greater than 0).
- This logarithm is undefined.
-
\(\log_{2.5}(6.25)\):
- Base \(b = 2.5\) (valid, as \(b > 0\) and \(b \neq 1\)).
- Argument \(x = 6.25\) (valid, as \(x > 0\)).
- This logarithm is defined.
-
\(\log_{1/3}(1/9)\):
- Base \(b = \frac{1}{3}\) (valid, as it is greater than 0 but less than 1).
- Argument \(x = \frac{1}{9}\) (valid, as \(x > 0\)).
- This logarithm is defined.
Based on the evaluations, the only logarithm that is undefined among the options presented is:
- \(\log_5(0)\).