Complete the table for the function [tex]$y=\log _{\frac{1}{3}}(x)+4$[/tex].
| x | y |
|---|---|
| [tex]$\frac{1}{9}$[/tex] | |
| [tex]$\frac{1}{3}$[/tex] | |
| 1 | |
| 3 | |
| 9 | |
Graph the function [tex]$y=\log _{\frac{1}{T}}(x)+4$[/tex]. Plot two points with integer coordinates to graph the function.
Answer (1)
Calculate y for x = 9 1 : y = lo g 3 1 ( 9 1 ) + 4 = 2 + 4 = 6 .
Calculate y for x = 3 1 : y = lo g 3 1 ( 3 1 ) + 4 = 1 + 4 = 5 .
Calculate y for x = 1 : y = lo g 3 1 ( 1 ) + 4 = 0 + 4 = 4 .
Calculate y for x = 3 : y = lo g 3 1 ( 3 ) + 4 = − 1 + 4 = 3 .
Calculate y for x = 9 : y = lo g 3 1 ( 9 ) + 4 = − 2 + 4 = 2 .
Two points with integer coordinates are ( 1 , 4 ) and ( 3 , 3 ) .
The completed table is:
x
y
9 1
6
3 1
5
1
4
3
3
9
2
.
Explanation
Understanding the Problem We are given the function y = lo g 3 1 ( x ) + 4 and asked to complete the table for the given values of x : 9 1 , 3 1 , 1 , 3 , 9 . We also need to graph the function and plot two points with integer coordinates.
Calculating y values First, let's calculate the y values for each given x value using the function y = lo g 3 1 ( x ) + 4 .
Calculating y for x=1/9 For x = 9 1 :
y = lo g 3 1 ( 9 1 ) + 4 Since ( 3 1 ) 2 = 9 1 , we have lo g 3 1 ( 9 1 ) = 2 . Therefore, y = 2 + 4 = 6
Calculating y for x=1/3 For x = 3 1 :
y = lo g 3 1 ( 3 1 ) + 4 Since ( 3 1 ) 1 = 3 1 , we have lo g 3 1 ( 3 1 ) = 1 . Therefore, y = 1 + 4 = 5
Calculating y for x=1 For x = 1 :
y = lo g 3 1 ( 1 ) + 4 Since ( 3 1 ) 0 = 1 , we have lo g 3 1 ( 1 ) = 0 . Therefore, y = 0 + 4 = 4
Calculating y for x=3 For x = 3 :
y = lo g 3 1 ( 3 ) + 4 Since ( 3 1 ) − 1 = 3 , we have lo g 3 1 ( 3 ) = − 1 . Therefore, y = − 1 + 4 = 3
Calculating y for x=9 For x = 9 :
y = lo g 3 1 ( 9 ) + 4 Since ( 3 1 ) − 2 = 9 , we have lo g 3 1 ( 9 ) = − 2 . Therefore, y = − 2 + 4 = 2
Completed Table Now we can complete the table:
x
y
9 1
6
3 1
5
1
4
3
3
9
2
Identifying Points for Graphing From the completed table, we can identify several points with integer coordinates. For example, ( 1 , 4 ) and ( 3 , 3 ) . We can use these two points to graph the function y = lo g 3 1 ( x ) + 4 .
Graphing the Function The graph of the function y = lo g 3 1 ( x ) + 4 passes through the points ( 1 , 4 ) and ( 3 , 3 ) . The function is a logarithmic function with a base less than 1, so it is a decreasing function.
Final Points for Graphing The two points with integer coordinates that we will use to graph the function are ( 1 , 4 ) and ( 3 , 3 ) .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes (Richter scale), the loudness of sounds (decibels), and the acidity of solutions (pH scale). The function y = lo g 3 1 ( x ) + 4 is a transformed logarithmic function, where the base is 3 1 and there is a vertical shift of 4 units. Understanding how to work with logarithmic functions and their transformations is essential in various scientific and engineering fields. For example, if you are designing an audio system, you might use logarithmic scales to represent sound intensity levels. Or, if you are studying population growth, you might use logarithmic functions to model the rate of growth over time.
