Given:
It is given that the function is [tex]f(x)=\log _{2}(4 x)[/tex]
We need to determine the average rate of change of the function over the interval x = 2 to x = 8.
Value of f(2):
Substituting x = 2 in the function, we get;
[tex]f(2)=\log _{2}(4 (2))[/tex]
[tex]f(2)=\log _{2}(8)[/tex]
[tex]f(2)=3[/tex]
Thus, the value of f(2) is 3.
Value of f(8):
Substituting x = 8 in the function, we get;
[tex]f(8)=\log _{2}(4 (8))[/tex]
[tex]f(8)=\log _{2}(32)[/tex]
[tex]f(8)=5[/tex]
Thus, the value of f(8) is 5.
Average rate of change:
The average rate of change can be determined using the formula,
[tex]Average=\frac{f(b)-f(a)}{b-a}[/tex]
Substituting a = 2 and b = 8 in the above formula, we get;
[tex]Average=\frac{f(8)-f(2)}{8-2}[/tex]
[tex]Average=\frac{5-3}{8-2}[/tex]
[tex]Average=\frac{2}{6}[/tex]
[tex]Average=\frac{1}{3}[/tex]
Thus, the average rate of change over the interval x = 2 to x = 8 is [tex]\frac{1}{3}[/tex]
