Which of the following is a logarithmic function?
A. [tex]$y=0.25 x$[/tex]
B. [tex]$y=x^{0.25}$[/tex]
C. [tex]$y=\log _{0.25} x$[/tex]
D. [tex]$y=(0.25)^x$[/tex]
Answer (2)
A logarithmic function is of the form y = lo g b x .
Examine each option and compare it to the general form.
y = lo g 0.25 x matches the form of a logarithmic function.
Therefore, the logarithmic function is y = lo g 0.25 x .
Explanation
Identifying the Functions We are given four functions:
y = 0.25 x
y = x 0.25
y = lo g 0.25 x
y = ( 0.25 ) x
We need to identify which one is a logarithmic function.
Analyzing Each Option A logarithmic function has the general form y = lo g b x , where b is a positive constant not equal to 1, and 0"> x > 0 .
Let's examine each option:
y = 0.25 x is a linear function because it's in the form y = m x , where m = 0.25 .
y = x 0.25 is a power function because the variable x is raised to a constant power.
y = lo g 0.25 x is a logarithmic function because it matches the general form y = lo g b x , where the base b = 0.25 .
y = ( 0.25 ) x is an exponential function because a constant (0.25) is raised to a variable power x .
Conclusion Based on the analysis, the logarithmic function is y = lo g 0.25 x .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the loudness of sound (decibels), and modeling population growth or radioactive decay. For example, the Richter scale uses logarithms to quantify the size of an earthquake, where each whole number increase on the scale represents a tenfold increase in amplitude. This allows us to easily compare the magnitudes of different earthquakes.
The logarithmic function from the provided options is y = lo g 0.25 x , as it matches the general form of a logarithmic function, which is y = lo g b x . All other options represent different kinds of functions. Thus, the correct choice is Option C.
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