Question

find the average rate of change of ( y = log_2 x ) over the interval ( 4 leq x leq 8 ). write your answer as a fraction in simplest form.
the average rate of change is

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 4\), \(b=8\), and \(f(x)=\log_{2}x\).

Step2: Calculate \(f(4)\) and \(f(8)\)

Using the property \(\log_{a}a^{k}=k\), for \(f(4)=\log_{2}4\), since \(4 = 2^{2}\), then \(f(4)=2\). For \(f(8)=\log_{2}8\), since \(8 = 2^{3}\), then \(f(8)=3\).

Step3: Substitute into the formula

Substitute \(f(4) = 2\), \(f(8)=3\), \(a = 4\), and \(b = 8\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{3 - 2}{8-4}=\frac{1}{4}\).

Answer:

\(\frac{1}{4}\)