3. Given $a=\log _{4} \mathrm{e}, b=\log _{3} 4, c=\log _{4} 5$, then the size relationship of $a, b, c$ is
$$
a \in(0,1), b, c>1 \text {, and } b-c=\frac{\ln 4}{\ln 3}-\frac{\ln 5}{\ln 4}=\frac{\ln ^{2} 4-\ln 3 \cdot \ln 5}{\ln 3 \cdot \ln 4} \text {, }
$$

while $\ln 3 \cdot \ln 5c$, so $b>c>a$.