Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To graph the function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex] and to plot two points with integer coordinates, let's follow these steps in detail:
### Step-by-Step Solution
1. Understanding the Function:
[tex]\[ y = -\log_{\frac{1}{6}}(x) \][/tex]
This equation transforms the logarithm from base [tex]\(\frac{1}{6}\)[/tex] by negating it, which affects the shape and direction of the graph.
2. Change of Base Formula:
To work with logarithms of any base, we use the change of base formula:
[tex]\[ \log_{\frac{1}{6}}(x) = \frac{\log(x)}{\log(\frac{1}{6})} \][/tex]
Given that [tex]\(\log(\frac{1}{6}) = \log(1) - \log(6) = 0 - \log(6) = -\log(6)\)[/tex], we can rewrite:
[tex]\[ \log_{\frac{1}{6}}(x) = \frac{\log(x)}{-\log(6)} = -\frac{\log(x)}{\log(6)} \][/tex]
Hence, our function becomes:
[tex]\[ y = -\left( -\frac{\log(x)}{\log(6)} \right) = \frac{\log(x)}{\log(6)} \][/tex]
3. Plotting the Graph:
We will plot the function:
[tex]\[ y = \frac{\log(x)}{\log(6)} \][/tex]
4. Calculate Specific Points:
To plot points with integer coordinates, we need specific [tex]\( x \)[/tex] values where both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are integers.
- First Point:
Let’s try [tex]\( x = 6 \)[/tex]:
[tex]\[ y = \frac{\log(6)}{\log(6)} = 1 \][/tex]
The first point is [tex]\( (6, 1) \)[/tex].
- Second Point:
Let’s try [tex]\( x = 36 \)[/tex]:
[tex]\[ y = \frac{\log(36)}{\log(6)} = \frac{2 \log(6)}{\log(6)} = 2 \][/tex]
The second point is [tex]\( (36, 2) \)[/tex], but since [tex]\( x = 36 \)[/tex] is not practical for visualization, we try another.
Alternatively, let’s use [tex]\( x = 1 \)[/tex]:
[tex]\[ y = \frac{\log(1)}{\log(6)} = \frac{0}{\log(6)} = 0 \][/tex]
The second point is [tex]\( (1, 0) \)[/tex].
### Final Graph
- Plot the Points:
[tex]\[ (6, 1) \quad \text{and} \quad (1, 0) \][/tex]
- Plot the Function:
The curve of [tex]\( y = \frac{\log(x)}{\log(6)} \)[/tex] (which is equivalent to the original function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex]) is shown on a standard [tex]\( xy \)[/tex]-coordinate plane.
- Shape and Direction:
Given the nature of logarithmic functions, particularly with this transformation, the graph will pass through the points and exhibit the typical increasing logarithmic behavior, rising more slowly as [tex]\( x \)[/tex] increases.
To finalize, you should plot the points [tex]\((6, 1)\)[/tex] and [tex]\((1, 0)\)[/tex] on the [tex]\( xy \)[/tex]-plane and draw a smooth curve through these points to represent the function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex]. The graph will start from [tex]\((1, 0)\)[/tex] and pass through [tex]\((6, 1)\)[/tex], continuing to increase gradually as [tex]\( x \)[/tex] increases.
### Step-by-Step Solution
1. Understanding the Function:
[tex]\[ y = -\log_{\frac{1}{6}}(x) \][/tex]
This equation transforms the logarithm from base [tex]\(\frac{1}{6}\)[/tex] by negating it, which affects the shape and direction of the graph.
2. Change of Base Formula:
To work with logarithms of any base, we use the change of base formula:
[tex]\[ \log_{\frac{1}{6}}(x) = \frac{\log(x)}{\log(\frac{1}{6})} \][/tex]
Given that [tex]\(\log(\frac{1}{6}) = \log(1) - \log(6) = 0 - \log(6) = -\log(6)\)[/tex], we can rewrite:
[tex]\[ \log_{\frac{1}{6}}(x) = \frac{\log(x)}{-\log(6)} = -\frac{\log(x)}{\log(6)} \][/tex]
Hence, our function becomes:
[tex]\[ y = -\left( -\frac{\log(x)}{\log(6)} \right) = \frac{\log(x)}{\log(6)} \][/tex]
3. Plotting the Graph:
We will plot the function:
[tex]\[ y = \frac{\log(x)}{\log(6)} \][/tex]
4. Calculate Specific Points:
To plot points with integer coordinates, we need specific [tex]\( x \)[/tex] values where both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are integers.
- First Point:
Let’s try [tex]\( x = 6 \)[/tex]:
[tex]\[ y = \frac{\log(6)}{\log(6)} = 1 \][/tex]
The first point is [tex]\( (6, 1) \)[/tex].
- Second Point:
Let’s try [tex]\( x = 36 \)[/tex]:
[tex]\[ y = \frac{\log(36)}{\log(6)} = \frac{2 \log(6)}{\log(6)} = 2 \][/tex]
The second point is [tex]\( (36, 2) \)[/tex], but since [tex]\( x = 36 \)[/tex] is not practical for visualization, we try another.
Alternatively, let’s use [tex]\( x = 1 \)[/tex]:
[tex]\[ y = \frac{\log(1)}{\log(6)} = \frac{0}{\log(6)} = 0 \][/tex]
The second point is [tex]\( (1, 0) \)[/tex].
### Final Graph
- Plot the Points:
[tex]\[ (6, 1) \quad \text{and} \quad (1, 0) \][/tex]
- Plot the Function:
The curve of [tex]\( y = \frac{\log(x)}{\log(6)} \)[/tex] (which is equivalent to the original function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex]) is shown on a standard [tex]\( xy \)[/tex]-coordinate plane.
- Shape and Direction:
Given the nature of logarithmic functions, particularly with this transformation, the graph will pass through the points and exhibit the typical increasing logarithmic behavior, rising more slowly as [tex]\( x \)[/tex] increases.
To finalize, you should plot the points [tex]\((6, 1)\)[/tex] and [tex]\((1, 0)\)[/tex] on the [tex]\( xy \)[/tex]-plane and draw a smooth curve through these points to represent the function [tex]\( y = -\log_{\frac{1}{6}}(x) \)[/tex]. The graph will start from [tex]\((1, 0)\)[/tex] and pass through [tex]\((6, 1)\)[/tex], continuing to increase gradually as [tex]\( x \)[/tex] increases.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
