Question
Graph the following logarithmic functions. Which has a graph that decreases from left to right?(1 point)
Responses
log2x
logarithm subscript 2 baseline x
log−2x
logarithm subscript negative 2 baseline x
log0.5x
logarithm subscript 0.5 baseline x
lnx
Responses
log2x
logarithm subscript 2 baseline x
log−2x
logarithm subscript negative 2 baseline x
log0.5x
logarithm subscript 0.5 baseline x
lnx
Answers
GPT-4o mini
To determine which of the given logarithmic functions has a graph that decreases from left to right, let's analyze each function:
1. **\( \log_2(x) \)**: This is the logarithm base 2. It is an increasing function, meaning that as \( x \) increases, \( \log_2(x) \) also increases.
2. **\( \log_{-2}(x) \)**: Logarithms with a negative base are not defined in the real number system because logarithms require a base that is positive and not equal to 1. Hence, this function does not have a valid graph.
3. **\( \log_{0.5}(x) \)**: This logarithm has a base smaller than 1 (0.5). Logarithmic functions with bases less than 1 are decreasing functions. Therefore, as \( x \) increases, \( \log_{0.5}(x) \) decreases.
4. **\( \ln(x) \)**: The natural logarithm is also an increasing function. As \( x \) increases, \( \ln(x) \) increases.
From the analysis:
- The function that decreases from left to right is **\( \log_{0.5}(x) \)**.
1. **\( \log_2(x) \)**: This is the logarithm base 2. It is an increasing function, meaning that as \( x \) increases, \( \log_2(x) \) also increases.
2. **\( \log_{-2}(x) \)**: Logarithms with a negative base are not defined in the real number system because logarithms require a base that is positive and not equal to 1. Hence, this function does not have a valid graph.
3. **\( \log_{0.5}(x) \)**: This logarithm has a base smaller than 1 (0.5). Logarithmic functions with bases less than 1 are decreasing functions. Therefore, as \( x \) increases, \( \log_{0.5}(x) \) decreases.
4. **\( \ln(x) \)**: The natural logarithm is also an increasing function. As \( x \) increases, \( \ln(x) \) increases.
From the analysis:
- The function that decreases from left to right is **\( \log_{0.5}(x) \)**.