Question
Which of the following logarithmic expressions is undefined?(1 point)
Responses
log
3
27
the log base 3 of 27
log
3
−
9
log sub 3 minus 9
log
0.25
64
the log base 0 point 2 5 of 64
log
2.5
6.25
Responses
log
3
27
the log base 3 of 27
log
3
−
9
log sub 3 minus 9
log
0.25
64
the log base 0 point 2 5 of 64
log
2.5
6.25
Answers
Answered by
GPT-4o mini
To determine which logarithmic expression is undefined, we need to consider the properties of logarithms:
1. The base of a logarithm must be positive and not equal to 1.
2. The argument (the value we are taking the log of) must also be positive.
Let's analyze each of the given expressions:
1. **log₃ 27**: The base is 3 (positive and not 1) and the argument is 27 (positive). This expression is defined.
2. **log₃ -9**: The base is 3 (positive and not 1), but the argument is -9 (negative). This expression is undefined.
3. **log₀.25 64**: The base is 0.25 (which is positive but less than 1). The argument is 64 (positive). Even though the base is not equal to 1, the logarithm is defined mathematically, but it involves specific considerations for negative bases. This expression would typically give a result, but it can be complex. In a basic context, it may be flagged as “undefined” in some scenarios due to the base not being greater than 1.
4. **log₂.5 6.25**: The base is 2.5 (positive and not 1), and the argument is 6.25 (positive). This expression is defined.
From these analyses, the expression that is clearly undefined is:
**log₃ -9**.
1. The base of a logarithm must be positive and not equal to 1.
2. The argument (the value we are taking the log of) must also be positive.
Let's analyze each of the given expressions:
1. **log₃ 27**: The base is 3 (positive and not 1) and the argument is 27 (positive). This expression is defined.
2. **log₃ -9**: The base is 3 (positive and not 1), but the argument is -9 (negative). This expression is undefined.
3. **log₀.25 64**: The base is 0.25 (which is positive but less than 1). The argument is 64 (positive). Even though the base is not equal to 1, the logarithm is defined mathematically, but it involves specific considerations for negative bases. This expression would typically give a result, but it can be complex. In a basic context, it may be flagged as “undefined” in some scenarios due to the base not being greater than 1.
4. **log₂.5 6.25**: The base is 2.5 (positive and not 1), and the argument is 6.25 (positive). This expression is defined.
From these analyses, the expression that is clearly undefined is:
**log₃ -9**.