Which table represents the graph of a logarithmic function in the form [tex]y=\log _b x[/tex] when [tex]b>1[/tex]?
Table 1:
| x | y |
| :----- | :---- |
| 1/8 | -3 |
| 1/4 | -2 |
| 1/2 | -1 |
| 1 | 0 |
| 2 | 1 |
Table 2:
| x | y |
| :------ | :------- |
| -1.9 | -2.096 |
| -1.75 | -1.262 |
Answer (2)
Logarithmic functions are only defined for positive x values.
The second table contains negative x values, so it cannot represent a logarithmic function.
The first table has positive x values. Assuming y = lo g b x , we find b = 2 using the point ( 8 1 , − 3 ) .
Verify that y = lo g 2 x holds for all points in the first table. Thus, the first table represents the logarithmic function. The first table represents the graph of a logarithmic function in the form y = lo g b x when 1"> b > 1 .
Explanation
Understanding Logarithmic Functions We are given two tables of x and y values and asked to identify which one represents a logarithmic function of the form y = lo g b x where 1"> b > 1 . Let's analyze the properties of logarithmic functions to help us determine the correct table.
Key Properties A logarithmic function y = lo g b x is only defined for positive values of x . This means any table with non-positive x values cannot represent a logarithmic function. Also, for 1"> b > 1 , the logarithmic function is increasing.
Analyzing the Second Table The second table has x values of − 1.9 and − 1.75 , which are both negative. Therefore, the second table cannot represent a logarithmic function.
Analyzing the First Table Now let's examine the first table. The x values are 8 1 , 4 1 , 2 1 , 1 , 2 , and the corresponding y values are − 3 , − 2 , − 1 , 0 , 1 . All x values are positive, so it could potentially represent a logarithmic function.
Finding the Base Let's assume the first table represents y = lo g b x . We can use one of the points to find the base b . For example, using the point ( 8 1 , − 3 ) , we have − 3 = lo g b 8 1 . This means b − 3 = 8 1 , so b = 2 .
Verifying the Base Now we need to verify that b = 2 works for all other points in the first table.
For x = 4 1 , y = lo g 2 4 1 = − 2 , which matches the table. For x = 2 1 , y = lo g 2 2 1 = − 1 , which matches the table. For x = 1 , y = lo g 2 1 = 0 , which matches the table. For x = 2 , y = lo g 2 2 = 1 , which matches the table.
Since all points in the first table satisfy y = lo g 2 x , the first table represents a logarithmic function with base 2.
Conclusion Therefore, the first table represents the graph of a logarithmic function in the form y = lo g b x when 1"> b > 1 .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, modeling population growth, and calculating the pH of a solution. Understanding logarithmic functions helps us analyze and interpret data in these areas. For example, the Richter scale uses logarithms to quantify the size of an earthquake. An earthquake of magnitude 7 is ten times larger than an earthquake of magnitude 6. This is because the Richter scale is logarithmic, meaning that each whole number increase represents a tenfold increase in amplitude.
The first table represents a logarithmic function because all x values are positive, allowing us to determine the base as b = 2 by verifying the corresponding y values using the logarithm function. The second table contains negative x values, which cannot represent a logarithmic function. Thus, the first table is the correct choice.
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