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:
- The base \( b \) must be greater than 0 and cannot be equal to 1 (i.e., \( b > 0 \) and \( b \neq 1 \)).
- The argument \( x \) must be greater than 0 (i.e., \( x > 0 \)).
Let’s evaluate each option:
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\( \log_{13} 19 \): Here, \( b = 13 \) (valid) and \( x = 19 \) (valid). This expression is defined.
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\( \log_{\frac{1}{3}} \frac{1}{9} \): 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.
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\( \log_{5} 1 \): The base \( b = 5 \) (valid), and the argument \( x = 1 \) (valid since \( x > 0 \)). This expression is defined.
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\( \log_{2.5} 6.25 \): The base \( b = 2.5 \) (valid), and the argument \( x = 6.25 \) (valid since \( x > 0 \)). This expression is defined.
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\( \log_{5} 0 \): 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:
\( \log_{5} 0 \).