Which properties are present in a table that represents a logarithmic function in the form [tex]$y=\log _0 x_{\text {when }} b\ \textgreater \ 1$[/tex]?
I. The [tex]$y$[/tex]-values are always increasing or always decreasing.
II. The point [tex]$(0,1)$[/tex] exists in the table.
III. The [tex]$y$[/tex]-values will decrease rapidly as the [tex]$x$[/tex]-values approach zero.
IV. There will only be one [tex]$x$[/tex]-value in the table with a [tex]$y$[/tex]-value of zero.
A. I only
B. I and II only
C. I, III, and IV
D. II and III only
Answer (2)
Logarithmic functions with 1"> b > 1 are increasing, so their y -values always increase.
The point ( 0 , 1 ) cannot exist in the table because the domain of the logarithmic function is 0"> x > 0 .
As x approaches 0, the y -values decrease rapidly towards negative infinity.
There is only one x -value (which is x = 1 ) where the y -value is zero, since lo g b 1 = 0 .
The correct answer is I, III, and IV, so the answer is I, III, and IV .
Explanation
Analyzing the Problem We are asked to identify the correct properties of a logarithmic function y = lo g b x when 1"> b > 1 from the given options. Let's analyze each statement.
Analyzing Statement I Statement I: The y -values are always increasing or always decreasing. For 1"> b > 1 , the logarithmic function is an increasing function. As x increases, y also increases. Therefore, the y -values are always increasing. This statement is TRUE.
Analyzing Statement II Statement II: The point ( 0 , 1 ) exists in the table. The domain of the logarithmic function y = lo g b x is 0"> x > 0 . Therefore, x cannot be 0. Also, lo g b 0 is undefined. Thus, the point ( 0 , 1 ) cannot exist in the table. This statement is FALSE.
Analyzing Statement III Statement III: The y -values will decrease rapidly as the x -values approach zero. As x approaches 0 from the right (since 0"> x > 0 ), the y -values approach negative infinity. This means the y -values decrease rapidly as x approaches zero. This statement is TRUE.
Analyzing Statement IV Statement IV: There will only be one x -value in the table with a y -value of zero. The y -value is zero when x = 1 , since lo g b 1 = 0 for any 0"> b > 0 and b = 1 . There is only one such x -value. This statement is TRUE.
Conclusion Therefore, statements I, III, and IV are true.
Examples
Logarithmic functions are used to model many real-world phenomena, such as the Richter scale for measuring the magnitude of earthquakes. The properties of logarithmic functions, such as their increasing or decreasing nature and their behavior as x approaches zero, are crucial in understanding and interpreting these models. For example, understanding how the y-values decrease rapidly as the x-values approach zero helps us to understand the intensity of an earthquake as we move closer to the epicenter.
The true properties for a logarithmic function with base 1"> b > 1 are that the y -values are always increasing (I), they decrease rapidly as x approaches zero (III), and there is only one x -value where the y -value is zero (IV). Thus, the correct choice is I, III, and IV. The answer is therefore I, III, and IV .
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