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][tex]$y=(0.25)^x$[/tex][/tex]



Answer :

To determine which of the given options is a logarithmic function, we need to understand the form of a logarithmic function. A logarithmic function can be generally expressed as [tex]\( y = \log_a x \)[/tex], where [tex]\( a \)[/tex] is the base of the logarithm.

Let's analyze each option:

1. [tex]\( y = 0.25x \)[/tex]: This is a linear function because the variable [tex]\( x \)[/tex] is to the first power and it has a constant coefficient of 0.25. It does not have the form of a logarithmic function.

2. [tex]\( y = x^{0.25} \)[/tex]: This is a power function. The exponent 0.25 is applied to the variable [tex]\( x \)[/tex]. It is not a logarithmic function.

3. [tex]\( y = \log_{0.25} x \)[/tex]: This is a logarithmic function because it is in the form [tex]\( y = \log_a x \)[/tex], with the base [tex]\( a \)[/tex] being 0.25.

4. [tex]\( y = (0.25)^x \)[/tex]: This is an exponential function because the base 0.25 is raised to the power of [tex]\( x \)[/tex]. It is not a logarithmic function.

Thus, the logarithmic function among the given options is:
[tex]\[ y = \log_{0.25} x \][/tex]

Therefore, the correct option is:
[tex]\[ y = \log_{0.25} x \][/tex]