15. Find the number of positive integers $x$ that satisfy the inequality $\left|3+\log _{x} \frac{1}{3}\right|<\frac{8}{3}$.
15. Answer: 25
We have
$$
\begin{array}{c}
\quad-\frac{8}{3}<3+\log _{x} \frac{1}{3}<\frac{8}{3} \\
-\frac{17}{3}<-\frac{1}{\log _{3} x}<-\frac{1}{3} \\
\frac{3}{17}<\log _{3} x<3 \\
3^{3 / 17}<x<27
\end{array}
$$

As $1<3^{3 / 1 \dagger}<2$, we.see that integer solutions of the last inequality are $x=2$, $3, \ldots, 26$. Hence there are 25 of them.