1. Solve the inequality

$$
\log 2^{2}+\log 3^{1+\frac{1}{2 x}}-\log \left(3^{\frac{1}{x}}+3^{3}\right)>0
$$
1. We rewrite the inequality in an equivalent form:

$$
\begin{aligned}
& \log \frac{2^{2} \cdot 3^{1+\frac{1}{2 x}}}{3^{\frac{1}{x}}+3^{3}}>\log 1 \\
\Rightarrow \quad & \frac{2^{2} \cdot 3^{1+\frac{1}{2 x}}}{3^{\frac{1}{x}}+3^{3}}>1 \\
3^{\frac{1}{x}}-12 \cdot 3^{\frac{1}{2 x}}+27 & 0$, we obtain the inequality

10 points

$$
t^{2}-12 t+27<0 \quad \text { i.e. } \quad(t-3)(t-9)<0
$$

whose solution is $t \in(3,9)$.

10 points

The solution to the given inequality is $x \in\left(\frac{1}{4}, \frac{1}{2}\right)$.

5 points