3. Given $\log _{a} b+3 \log _{b} a=\frac{13}{2}$, when $a>b>1$, the value of $\frac{a+b^{4}}{a^{2}+b^{2}}$ is
A. 13
B. 4
C. 2
D. 1
Solve:
$\log _{a} b+\frac{3}{\log _{a} b}=\frac{13}{2} \Rightarrow \log _{a} b=\frac{1}{2}$ or $\log _{a} b=6$ (discard). Thus, $b=\sqrt{a} \Rightarrow \frac{a+b^{4}}{a^{2}+b^{2}}=\frac{a+a^{2}}{a+a^{2}}=1$, so the answer is $D$.