5. If $a=\frac{13}{6}, b=\log _{3} 10, c=\log _{4} 20$, then the size relationship between $a 、 b 、 c$ is $\qquad$ .
5. $a>c>b$.

Notice that $a>c \Leftrightarrow \frac{13}{6}>\log _{4} 20 \Leftrightarrow 4^{\frac{13}{6}}>20 \Leftrightarrow 4^{\frac{1}{6}}>\frac{5}{4}$ $\Leftrightarrow 2^{\frac{1}{3}}>\frac{5}{4} \Leftrightarrow 2>\frac{5^{3}}{4^{3}}=\frac{125}{64} \Leftrightarrow 2 \times 64=128>125$.
Below we prove $b10 \\
\Leftrightarrow 3^{\frac{1}{7}}>\frac{10}{9} \Leftrightarrow 3>\frac{10^{7}}{9^{7}}=\frac{10000000}{4782969} \\
\Leftrightarrow 3 \times 4782969>10^{7}, \\
c>\frac{15}{7} \Leftrightarrow \log _{4} 20>\frac{15}{7} \Leftrightarrow 4^{\frac{15}{7}}c>b$.