4. Let $f(x)=|\ln x|$, if the function $g(x)=f(x)-a x$ has three zeros in the interval $(0,4)$, then the range of the real number $a$ is $\qquad$ .
Let $y=a x$ be tangent to $y=\ln x$ at point $A\left(x_{0}, \ln x_{0}\right)$, then $y-\ln x_{0}=\frac{1}{x_{0}}\left(x-x_{0}\right)$ $\Rightarrow y=\frac{1}{x_{0}} x-1+\ln x_{0} \Rightarrow x_{0}=e$. Combining with the graph, the range of the real number $a$ is $\left(\frac{\ln 2}{2}, \frac{1}{e}\right)$.