Which of the following is a logarithmic function?
$y=0.25 x$
$y=x^{0.25}$
$y=\log _{0.25} x$
$y=(0.25)^x$
Answer (1)
Identify the general form of a logarithmic function as y = lo g b x .
Examine each given function and determine its type.
y = 0.25 x is a linear function.
y = x 0.25 is a power function.
y = lo g 0.25 x is a logarithmic function.
y = ( 0.25 ) x is an exponential function.
The logarithmic function is y = lo g 0.25 x .
Explanation
Identifying Logarithmic Functions We are given four functions and we need to identify the logarithmic function among them. A logarithmic function has the general form y = lo g b x , where b is the base and x is the argument.
Analyzing Each Function Let's examine each function:
y = 0.25 x is a linear function because it is 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 b = 0.25 .
y = ( 0.25 ) x is an exponential function because a constant (0.25) is raised to a variable power x .
Conclusion Therefore, the logarithmic function among the given options 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. 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. Understanding logarithmic functions helps us interpret and analyze these phenomena effectively.
