The function shown in the given graph is [tex]\rm f(x) = log_5(x)[/tex] and this can be determined by substituting the value of 'x' in the given options.
Given :
Points on the given graph of logarithmic function -- (0.2,-1), (1,0), and (5,1).
The following steps can be used in order to determine the function that shown in the graph:
Step 1 - Check option A) by substituting the value of x = {0.2,1,5}.
[tex]\rm f(0.2) = log_{0.2}(0.2) = 1[/tex]
[tex]\rm f(1) = log_{0.2}(1) = 0[/tex]
[tex]\rm f(5) = log_{0.2}(5) = -1[/tex]
Therefore, this option is incorrect.
Step 2 - Check option B) by substituting the value of x = {0.2,1,5}.
[tex]\rm f(0.2) = log_{2}(0.2) = -2.32[/tex]
[tex]\rm f(1) = log_{2}(1) = 0[/tex]
[tex]\rm f(5) = log_{2}(5) = 2.32[/tex]
Therefore, this option is incorrect.
Step 3 - Check option C) by substituting the value of x = {0.2,1,5}.
[tex]\rm f(0.2) = log_{5}(0.2) = -1[/tex]
[tex]\rm f(1) = log_{5}(1) = 0[/tex]
[tex]\rm f(5) = log_{5}(5) = 1[/tex]
Therefore, this option is correct.
For more information, refer to the link given below:
https://brainly.com/question/20381610
