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 decreases from left to right, let's analyze each function:
1. **log₂(x)**: This is the logarithm base 2. The graph of this function increases as \( x \) increases. For positive values of \( x \), it is always above the x-axis and becomes larger as \( x \) grows.
2. **log₋₂(x)**: Logarithms with a negative base are not defined in the real number system, so this function does not have a valid graph.
3. **log₀.₅(x)**: This is the logarithm base 0.5. Since 0.5 is less than 1, the graph of this function decreases as \( x \) increases. As \( x \) approaches 0, log₀.₅(x) approaches infinity, and as \( x \) increases, its value decreases.
4. **ln(x)**: This is the natural logarithm (base e). The graph of this function also increases as \( x \) increases. For positive values of \( x \), it is always above the x-axis, gradually increasing without bound.
Based on the analysis:
- The function that **decreases from left to right** is **log₀.₅(x)**.
1. **log₂(x)**: This is the logarithm base 2. The graph of this function increases as \( x \) increases. For positive values of \( x \), it is always above the x-axis and becomes larger as \( x \) grows.
2. **log₋₂(x)**: Logarithms with a negative base are not defined in the real number system, so this function does not have a valid graph.
3. **log₀.₅(x)**: This is the logarithm base 0.5. Since 0.5 is less than 1, the graph of this function decreases as \( x \) increases. As \( x \) approaches 0, log₀.₅(x) approaches infinity, and as \( x \) increases, its value decreases.
4. **ln(x)**: This is the natural logarithm (base e). The graph of this function also increases as \( x \) increases. For positive values of \( x \), it is always above the x-axis, gradually increasing without bound.
Based on the analysis:
- The function that **decreases from left to right** is **log₀.₅(x)**.