3. If $f(x)=\log _{a}\left(x+\frac{10 a}{x}\right)$ is a monotonic function on $[3,4]$, then the range of values for $a$ is $\qquad$
3. $\left(0, \frac{9}{10}\right] \cup\left[\frac{8}{5},+\infty\right)$.

Assume $f(x)$ is monotonically increasing on $[3,4]$.
If $a>1$, then $g(x)=x+\frac{10 a}{x}$ is also monotonically increasing on $[3,4]$. The minimum value of $g(x)$ is obtained when $x=\frac{10 a}{x}$, i.e., $x=\sqrt{10 a}$. Therefore, $\sqrt{10 a} \leqslant 3 \Rightarrow a \leqslant \frac{9}{10}$, which is a contradiction.

If $01$, similar discussion yields $\sqrt{10 a} \geqslant 4 \Rightarrow a \geqslant \frac{8}{5}$.
If $0<a<1$, similar discussion yields $\sqrt{10 a} \leqslant 3 \Rightarrow a \leqslant \frac{9}{10}$.
Thus, the range of $a$ is $\left(0, \frac{9}{10}\right] \cup\left[\frac{8}{5},+\infty\right)$.