Question
Use the graph of y = 5 Superscript x Baseline to answer the question.
Information
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens up and to the left and is labeled y equals 5 superscript x baseline. The curve approaches a horizontal asymptote at y equals 0 in quadrant 2, crosses the y-axis, and continues to rise in quadrant 1.
Question
Which graph shows y equals Log base 5 of x EndLog?
Answer options with 4 options
A.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens up and to the right. The curve starts in quadrant 2, crosses the y-axis, and approaches a horizontal asymptote at y equals 0 in quadrant 1.
C.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens up and to the right. The curve approaches a vertical asymptote at x equals 0 in quadrant 1, crosses the x-axis, and continues to decline slowly in quadrant 4.
B.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens down and to the right. The curve approaches a vertical asymptote at x equals 0 in quadrant 4, crosses the x-axis, and continues to rise slowly in quadrant 1.
D.
A graph.
Information
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens up and to the left and is labeled y equals 5 superscript x baseline. The curve approaches a horizontal asymptote at y equals 0 in quadrant 2, crosses the y-axis, and continues to rise in quadrant 1.
Question
Which graph shows y equals Log base 5 of x EndLog?
Answer options with 4 options
A.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens up and to the right. The curve starts in quadrant 2, crosses the y-axis, and approaches a horizontal asymptote at y equals 0 in quadrant 1.
C.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens up and to the right. The curve approaches a vertical asymptote at x equals 0 in quadrant 1, crosses the x-axis, and continues to decline slowly in quadrant 4.
B.
A graph.Short description, A graph.,Long description,
On a coordinate plane, a curve opens down and to the right. The curve approaches a vertical asymptote at x equals 0 in quadrant 4, crosses the x-axis, and continues to rise slowly in quadrant 1.
D.
A graph.
Answers
GPT-4o mini
To determine the correct representation of the graph for \( y = \log_5(x) \), we need to consider the characteristics of logarithmic functions.
1. The **domain** of \( y = \log_5(x) \) is \( x > 0 \), which means it only exists in the first quadrant and part of the fourth quadrant.
2. The **range** of \( y = \log_5(x) \) is all real numbers, which indicates it can take any vertical value.
3. As \( x \) approaches 0 from the right, \( y \) approaches negative infinity, creating a vertical asymptote at \( x = 0 \).
4. The graph passes through the point (1, 0), since \( \log_5(1) = 0 \).
Now, let's analyze the options provided:
- **Option A**: Describes a curve that starts in the second quadrant, crosses the y-axis, and approaches a horizontal asymptote at \( y = 0 \) in quadrant 1. This does not match \( y = \log_5(x) \).
- **Option B**: This option describes a curve that opens down and to the right, which is not a characteristic of logarithmic functions.
- **Option C**: This describes a curve that starts in quadrant 1, approaches a vertical asymptote at \( x = 0 \), and crosses the x-axis, which aligns with the behavior of a logarithm.
- **Option D**: There's no description, so we can't analyze this option.
Based on this analysis, the correct choice showing \( y = \log_5(x) \) is:
**C. A graph that opens up to the right, approaches a vertical asymptote at \( x = 0 \) in quadrant 1, crosses the x-axis, and continues to decline slowly in quadrant 4.**
1. The **domain** of \( y = \log_5(x) \) is \( x > 0 \), which means it only exists in the first quadrant and part of the fourth quadrant.
2. The **range** of \( y = \log_5(x) \) is all real numbers, which indicates it can take any vertical value.
3. As \( x \) approaches 0 from the right, \( y \) approaches negative infinity, creating a vertical asymptote at \( x = 0 \).
4. The graph passes through the point (1, 0), since \( \log_5(1) = 0 \).
Now, let's analyze the options provided:
- **Option A**: Describes a curve that starts in the second quadrant, crosses the y-axis, and approaches a horizontal asymptote at \( y = 0 \) in quadrant 1. This does not match \( y = \log_5(x) \).
- **Option B**: This option describes a curve that opens down and to the right, which is not a characteristic of logarithmic functions.
- **Option C**: This describes a curve that starts in quadrant 1, approaches a vertical asymptote at \( x = 0 \), and crosses the x-axis, which aligns with the behavior of a logarithm.
- **Option D**: There's no description, so we can't analyze this option.
Based on this analysis, the correct choice showing \( y = \log_5(x) \) is:
**C. A graph that opens up to the right, approaches a vertical asymptote at \( x = 0 \) in quadrant 1, crosses the x-axis, and continues to decline slowly in quadrant 4.**