Which table represents the graph of a logarithmic function in the form $y=\log _b x$ when $b>1$?
Answer (1)
Logarithmic functions are only defined for positive x values.
The first table has positive x values, while the second table has negative x values, so the second table cannot represent a logarithmic function.
The first table contains the point ( 1 , 0 ) , consistent with lo g b 1 = 0 .
The first table represents the logarithmic function y = lo g 2 x .
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Explanation
Understanding Logarithmic Functions We are given two tables of x and y values and asked to determine 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 identify the correct table.
Domain of Logarithmic Functions A logarithmic function y = lo g b x is only defined for positive values of x . This means that any table containing non-positive x values cannot represent a logarithmic function.
Analyzing the Tables Looking at the first table, all x values are positive: 8 1 , 4 1 , 2 1 , 1 , 2 . The second table contains negative x values: − 1.9 , − 1.75 . Therefore, the second table cannot represent a logarithmic function.
Checking the Base The first table contains the point ( 1 , 0 ) , which is consistent with the property that lo g b 1 = 0 for any base 0"> b > 0 . Let's check if we can find a base 1"> b > 1 that satisfies the other points in the table.
Verifying the Base Using the point ( 2 , 1 ) from the first table, we have 1 = lo g b 2 . This implies that b 1 = 2 , so b = 2 . Since 1"> 2 > 1 , this is a valid base. Let's check if the other points in the first table also satisfy the equation y = lo g 2 x .
Confirming the Function For x = 8 1 , y = lo g 2 8 1 = lo g 2 2 − 3 = − 3 . This matches the table. For x = 4 1 , y = lo g 2 4 1 = lo g 2 2 − 2 = − 2 . This matches the table. For x = 2 1 , y = lo g 2 2 1 = lo g 2 2 − 1 = − 1 . This matches the table. For x = 1 , y = lo g 2 1 = 0 . This matches the table. For x = 2 , y = lo g 2 2 = 1 . This matches the table.
Conclusion Therefore, the first table represents the logarithmic function y = lo g 2 x .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. Understanding logarithmic functions helps us analyze and interpret data in these fields. For example, the Richter scale uses logarithms to quantify the size of earthquakes, where each whole number increase represents a tenfold increase in amplitude. This allows scientists to compare the relative magnitudes of different earthquakes effectively.
