Answer
\[x=\frac{3}{\ln 4}\]
Work Step by Step
\[\begin{align}
& f\left( x \right)=\frac{1}{x}\text{ on }\left[ 1,4 \right] \\
& \text{The average is given by} \\
& {{f}_{avg}}=\frac{1}{b-a}\int_{a}^{b}{f\left( x \right)dx} \\
& \text{Therefore,} \\
& {{f}_{avg}}=\frac{1}{4-1}\int_{1}^{4}{\frac{1}{x}dx} \\
& {{f}_{avg}}=\frac{1}{3}\int_{1}^{4}{\frac{1}{x}}dx \\
& \text{Integrating} \\
& {{f}_{avg}}=\frac{1}{3}\left[ \ln x \right]_{1}^{4} \\
& {{f}_{avg}}=\frac{1}{3}\left[ \ln 4-\ln 1 \right] \\
& {{f}_{avg}}=\frac{\ln 4}{3} \\
& \text{The point at which the given function equals its average value is} \\
& \frac{1}{x}=\frac{\ln 4}{3} \\
& \text{Solve for }x \\
& x=\frac{3}{\ln 4} \\
\end{align}\]
