Answer
$\frac{2\ln2}{3}$
Work Step by Step
The average value of $f$ over $[1,4]$ is:
$$\frac{1}{4-1}\int_{1}^{4}\frac{1}{x}dx$$
$$\frac{1}{3}\int_{1}^{4}\frac{1}{x}dx$$
$$\frac{1}{3}[\ln x]_{1}^{4}=\frac{1}{3}(\ln 4-\ln 1)=\frac{\ln4}{3}=\frac{2\ln2}{3}$$