Complete the table for the function [tex]$y=\log (x)+3$[/tex].
| x | y |
|---|---|
| [tex]$\frac{1}{100}$[/tex] | |
| [tex]$\frac{1}{10}$[/tex] | |
| 1 | |
| 10 | |
Graph the function [tex]$y=\log (x)+3$[/tex]. Plot two points with integer coordinates to graph the function.
Answer (2)
Calculate y for x = 100 1 : y = lo g ( 100 1 ) + 3 = 1 .
Calculate y for x = 10 1 : y = lo g ( 10 1 ) + 3 = 2 .
Calculate y for x = 1 : y = lo g ( 1 ) + 3 = 3 .
Calculate y for x = 10 : y = lo g ( 10 ) + 3 = 4 . The two points with integer coordinates are (1, 3) and (10, 4). The completed table is shown above. See table and points above.
Explanation
Understanding the Problem We are given the function y = lo g ( x ) + 3 . We need to complete the table for x = 100 1 , 10 1 , 1 , 10 . Then, we need to graph the function y = lo g ( x ) + 3 and plot two points with integer coordinates to graph the function.
Calculating y values First, let's calculate the y values for each given x value using the function y = lo g ( x ) + 3 .
Calculating y for x=1/100 For x = 100 1 : y = lo g ( 100 1 ) + 3 = lo g ( 1 0 − 2 ) + 3 = − 2 + 3 = 1
Calculating y for x=1/10 For x = 10 1 : y = lo g ( 10 1 ) + 3 = lo g ( 1 0 − 1 ) + 3 = − 1 + 3 = 2
Calculating y for x=1 For x = 1 : y = lo g ( 1 ) + 3 = 0 + 3 = 3
Calculating y for x=10 For x = 10 : y = lo g ( 10 ) + 3 = 1 + 3 = 4
Completed Table and Points Now we have the completed table:
x
y
1/100
1
1/10
2
1
3
10
4
We can plot the points ( 100 1 , 1 ) , ( 10 1 , 2 ) , ( 1 , 3 ) , and ( 10 , 4 ) on the graph.
Choosing Points for Graphing We can choose two points with integer coordinates from the calculated points, such as ( 1 , 3 ) and ( 10 , 4 ) . These points can be used to graph the function y = lo g ( x ) + 3 .
Final Answer The final completed table is:
x
y
1/100
1
1/10
2
1
3
10
4
Two points with integer coordinates are (1, 3) and (10, 4).
Examples
Logarithmic functions are incredibly useful in many real-world scenarios. For instance, the Richter scale, which measures the magnitude of earthquakes, is a logarithmic scale. This means that an earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. Similarly, in chemistry, pH values are based on a logarithmic scale to measure the acidity or alkalinity of a solution. Understanding logarithmic functions helps scientists and engineers make sense of phenomena that span many orders of magnitude.
The completed table for the function y = lo g ( x ) + 3 is provided, with values calculated for x = 100 1 , 10 1 , 1 , and 10 . The points with integer coordinates suitable for plotting the graph are (1, 3) and (10, 4).
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