Which properties are present in a table that represents a logarithmic function in the form [tex]$y=\log _b x_{\text {when }} b\ \textgreater \ 1$[/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 (1)
Logarithmic functions of the form y = lo g b x where 1"> b > 1 are always increasing.
The point ( 0 , 1 ) does not exist in the table of values for the function.
The y -values decrease rapidly as the x -values approach zero.
There is only one x -value with a y -value of zero, which is x = 1 .
Therefore, the properties I, III, and IV are present. The answer is I , III , an d I V .
Explanation
Analyzing the Problem We are given a logarithmic function in the form y = lo g b x where 1"> b > 1 . We need to determine which of the given properties are present in a table representing this function. Let's analyze each property individually.
Analyzing Property I I. The y -values are always increasing or always decreasing. For 1"> b > 1 , the logarithmic function y = lo g b x is an increasing function. This means as x increases, y also increases. Therefore, the y -values are always increasing. So, property I is present.
Analyzing Property II II. The point ( 0 , 1 ) exists in the table. Let's check if the point ( 0 , 1 ) exists in the table. If x = 0 , then y = lo g b 0 , which is undefined. Also, the domain of the logarithmic function is 0"> x > 0 . Therefore, the point ( 0 , 1 ) does not exist in the table. So, property II is not present.
Analyzing Property III III. The y -values will decrease rapidly as the x -values approach zero. As x approaches zero from the right, i.e., x → 0 + , the y -values approach − ∞ . This means the y -values decrease rapidly as the x -values approach zero. So, property III is present.
Analyzing Property IV IV. There will only be one x -value in the table with a y -value of zero. Let's find the x -value when y = 0 . If y = 0 , then 0 = lo g b x . This implies x = b 0 = 1 . Therefore, there is only one x -value (which is x = 1 ) with a y -value of zero. So, property IV is present.
Conclusion Based on the analysis, properties I, III, and IV are present. Therefore, the correct answer is I, III, and IV.
Examples
Logarithmic functions are incredibly useful in many real-world scenarios. For instance, they are used to measure the intensity of earthquakes on the Richter scale. The scale is logarithmic because the intensity of earthquakes can vary over a huge range, and using a logarithmic scale allows us to represent these intensities in a more manageable way. Similarly, logarithmic scales are used in acoustics to measure sound intensity (decibels) and in chemistry to measure the acidity or alkalinity of a solution (pH). Understanding the properties of logarithmic functions helps us interpret and work with these scales effectively.
