3. Given $a=\log _{4} \mathrm{e}, b=\log _{3} 4, c=\log _{4} 5$. Then the size relationship of $a, b, c$ is $\qquad$ .
3. $a<c<b$.

Notice that, $a<1$,
$$
\begin{array}{l}
\frac{\log _{4} 5}{\log _{3} 4}=\log _{4} 5 \cdot \log _{4} 3 \\
<\left(\frac{\log _{4} 5+\log _{4} 3}{2}\right)^{2}=\left(\frac{\log _{4} 15}{2}\right)^{2} \\
<\left(\frac{\log _{4} 16}{2}\right)^{2}=1 .
\end{array}
$$

Therefore, $a<c<b$.