To determine which of the given functions is a logarithmic function, let's briefly review the characteristics of each type of function listed in the question.
1. Linear Function: A linear function has the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants.
- The function [tex]\( y = 0.25x \)[/tex] is a linear function because it can be written in the form [tex]\( y = mx + b \)[/tex] with [tex]\( m = 0.25 \)[/tex] and [tex]\( b = 0 \)[/tex]. Therefore, [tex]\( y = 0.25x \)[/tex] is a linear function.
2. Power Function: A power function has the form [tex]\( y = x^c \)[/tex], where [tex]\( c \)[/tex] is a constant.
- The function [tex]\( y = x^{0.25} \)[/tex] is a power function because it can be written in the form [tex]\( y = x^c \)[/tex] with [tex]\( c = 0.25 \)[/tex]. Therefore, [tex]\( y = x^{0.25} \)[/tex] is a power function.
3. Logarithmic Function: A logarithmic function has the form [tex]\( y = \log_b x \)[/tex], where [tex]\( b \)[/tex] is the base of the logarithm.
- The function [tex]\( y = \log_{0.25} x \)[/tex] is a logarithmic function because it fits the form [tex]\( y = \log_b x \)[/tex] with [tex]\( b = 0.25 \)[/tex]. Therefore, [tex]\( y = \log_{0.25} x \)[/tex] is a logarithmic function.
4. Exponential Function: An exponential function has the form [tex]\( y = b^x \)[/tex], where [tex]\( b \)[/tex] is a constant base and [tex]\( x \)[/tex] is the exponent.
- The function [tex]\( y = (0.25)^x \)[/tex] is an exponential function because it can be written in the form [tex]\( y = b^x \)[/tex] with [tex]\( b = 0.25 \)[/tex]. Therefore, [tex]\( y = (0.25)^x \)[/tex] is an exponential function.
From the above analysis, the function that fits the form of a logarithmic function is:
[tex]\[ y = \log_{0.25} x \][/tex]
So, the correct answer is:
[tex]\[ y = \log_{0.25} x \][/tex]
Among the given choices:
- [tex]\( y = 0.25 x \)[/tex] is a linear function.
- [tex]\( y = x^{0.25} \)[/tex] is a power function.
- [tex]\( y = \log_{0.25} x \)[/tex] is a logarithmic function.
- [tex]\( y = (0.25)^x \)[/tex] is an exponential function.
Therefore, the correct (logarithmic function) answer is [tex]\( y = \log_{0.25} x \)[/tex].
