Answer :
To graph the function [tex]\( y = -\log_{\frac{1}{7}}(x) \)[/tex], let's break down the necessary steps and identify and plot two points with integer coordinates.
### Understanding the Function
First, let's rewrite the function using a more convenient logarithmic base transformation. The base [tex]\( \frac{1}{7} \)[/tex] can be expressed as:
[tex]\[ y = -\log_{\frac{1}{7}}(x) \][/tex]
Using the change of base formula for logarithms, we can convert [tex]\( \log_{\frac{1}{7}}(x) \)[/tex] to a natural logarithm or any other base that's convenient. For simplicity, we will use the natural logarithm (base [tex]\( e \)[/tex]):
[tex]\[ \log_{\frac{1}{7}}(x) = \frac{\log_e(x)}{\log_e(\frac{1}{7})} \][/tex]
Thus,
[tex]\[ y = -\left( \frac{\log_e(x)}{\log_e(\frac{1}{7})} \right) \][/tex]
Since [tex]\( \log_e(\frac{1}{7}) \)[/tex] is negative (because [tex]\( \frac{1}{7} < 1 \)[/tex]):
[tex]\[ \log_e(\frac{1}{7}) = \log_e(1) - \log_e(7) = 0 - \log_e(7) = -\log_e(7) \][/tex]
So,
[tex]\[ y = -\left( \frac{\log_e(x)}{-\log_e(7)} \right) = \frac{\log_e(x)}{\log_e(7)} \][/tex]
Therefore, the function simplifies to:
[tex]\[ y = \frac{\log_e(x)}{\log_e(7)} \][/tex]
Since [tex]\( \log_e(7) \)[/tex] is a constant, this function describes a logarithmic curve.
### Plotting Points with Integer Coordinates
We need to find two points with integer coordinates that lie on the curve [tex]\( y = \frac{\log_e(x)}{\log_e(7)} \)[/tex]. Let's choose some simple values for [tex]\( x \)[/tex].
1. First Point:
Let [tex]\( x = 1 \)[/tex]:
[tex]\[ y = \frac{\log_e(1)}{\log_e(7)} = \frac{0}{\log_e(7)} = 0 \][/tex]
So, the point [tex]\( (1, 0) \)[/tex] lies on the graph.
2. Second Point:
Let [tex]\( x = 7 \)[/tex]:
[tex]\[ y = \frac{\log_e(7)}{\log_e(7)} = 1 \][/tex]
So, the point [tex]\( (7, 1) \)[/tex] lies on the graph.
### Graphing the Function
1. Plot the points [tex]\( (1, 0) \)[/tex] and [tex]\( (7, 1) \)[/tex] on the Cartesian plane.
2. Draw the curve:
- The function [tex]\( y = \frac{\log_e(x)}{\log_e(7)} \)[/tex] is increasing and passes through these points.
- The curve will start from the x-axis (since [tex]\( \log_e(1) = 0 \)[/tex]) and pass through [tex]\( (7, 1) \)[/tex].
- As [tex]\( x \)[/tex] approaches 0 from the positive side, [tex]\( y \)[/tex] becomes large negative because [tex]\( \log_e(x) \to -\infty \)[/tex].
### Summary of Points:
- The points [tex]\( (1, 0) \)[/tex] and [tex]\( (7, 1) \)[/tex] are easy to calculate and integer-valued points on the graph of the function [tex]\( y = -\log_{\frac{1}{7}}(x) \)[/tex].
By plotting these points and smoothly connecting them while following the behavior of the logarithmic function, you can accurately sketch the graph of [tex]\( y = -\log_{\frac{1}{7}}(x) \)[/tex].
### Understanding the Function
First, let's rewrite the function using a more convenient logarithmic base transformation. The base [tex]\( \frac{1}{7} \)[/tex] can be expressed as:
[tex]\[ y = -\log_{\frac{1}{7}}(x) \][/tex]
Using the change of base formula for logarithms, we can convert [tex]\( \log_{\frac{1}{7}}(x) \)[/tex] to a natural logarithm or any other base that's convenient. For simplicity, we will use the natural logarithm (base [tex]\( e \)[/tex]):
[tex]\[ \log_{\frac{1}{7}}(x) = \frac{\log_e(x)}{\log_e(\frac{1}{7})} \][/tex]
Thus,
[tex]\[ y = -\left( \frac{\log_e(x)}{\log_e(\frac{1}{7})} \right) \][/tex]
Since [tex]\( \log_e(\frac{1}{7}) \)[/tex] is negative (because [tex]\( \frac{1}{7} < 1 \)[/tex]):
[tex]\[ \log_e(\frac{1}{7}) = \log_e(1) - \log_e(7) = 0 - \log_e(7) = -\log_e(7) \][/tex]
So,
[tex]\[ y = -\left( \frac{\log_e(x)}{-\log_e(7)} \right) = \frac{\log_e(x)}{\log_e(7)} \][/tex]
Therefore, the function simplifies to:
[tex]\[ y = \frac{\log_e(x)}{\log_e(7)} \][/tex]
Since [tex]\( \log_e(7) \)[/tex] is a constant, this function describes a logarithmic curve.
### Plotting Points with Integer Coordinates
We need to find two points with integer coordinates that lie on the curve [tex]\( y = \frac{\log_e(x)}{\log_e(7)} \)[/tex]. Let's choose some simple values for [tex]\( x \)[/tex].
1. First Point:
Let [tex]\( x = 1 \)[/tex]:
[tex]\[ y = \frac{\log_e(1)}{\log_e(7)} = \frac{0}{\log_e(7)} = 0 \][/tex]
So, the point [tex]\( (1, 0) \)[/tex] lies on the graph.
2. Second Point:
Let [tex]\( x = 7 \)[/tex]:
[tex]\[ y = \frac{\log_e(7)}{\log_e(7)} = 1 \][/tex]
So, the point [tex]\( (7, 1) \)[/tex] lies on the graph.
### Graphing the Function
1. Plot the points [tex]\( (1, 0) \)[/tex] and [tex]\( (7, 1) \)[/tex] on the Cartesian plane.
2. Draw the curve:
- The function [tex]\( y = \frac{\log_e(x)}{\log_e(7)} \)[/tex] is increasing and passes through these points.
- The curve will start from the x-axis (since [tex]\( \log_e(1) = 0 \)[/tex]) and pass through [tex]\( (7, 1) \)[/tex].
- As [tex]\( x \)[/tex] approaches 0 from the positive side, [tex]\( y \)[/tex] becomes large negative because [tex]\( \log_e(x) \to -\infty \)[/tex].
### Summary of Points:
- The points [tex]\( (1, 0) \)[/tex] and [tex]\( (7, 1) \)[/tex] are easy to calculate and integer-valued points on the graph of the function [tex]\( y = -\log_{\frac{1}{7}}(x) \)[/tex].
By plotting these points and smoothly connecting them while following the behavior of the logarithmic function, you can accurately sketch the graph of [tex]\( y = -\log_{\frac{1}{7}}(x) \)[/tex].
