Question

find the average rate of change of ( y = log_3 x ) over the interval ( 1 leq x leq 3 ). write your answer as a fraction in simplest form. the average rate of change is ( square ).

Explanation:

Step1: Recall the formula for average rate of change

The formula for 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, \(a = 1\), \(b=3\), and \(f(x)=\log_{3}x\).

Step2: Calculate \(f(1)\) and \(f(3)\)

• For \(x = 1\): \(f(1)=\log_{3}1\). Since \(3^{0}=1\), then \(\log_{3}1 = 0\).
• For \(x = 3\): \(f(3)=\log_{3}3\). Since \(3^{1}=3\), then \(\log_{3}3=1\).

Step3: Substitute into the formula

Substitute \(f(1) = 0\), \(f(3)=1\), \(a = 1\), and \(b = 3\) into \(\frac{f(b)-f(a)}{b - a}\). We get \(\frac{1-0}{3 - 1}=\frac{1}{2}\).

Answer:

\(\frac{1}{2}\)