1A. In the set of real numbers, solve the equation

$$
\log (\log x)+\log \left(\log x^{3}-2\right)=0
$$
Solution. First, let's note that it must hold that $x>0, \log x>0$ and $\log x^{3}-2>0$, from which we get that $\log x>\frac{2}{3}$, or $x>\sqrt[3]{100}$.

We have:

$$
\begin{gathered}
\log (\log x)+\log \left(\log x^{3}-2\right)=0 \Leftrightarrow \log \left(\log x\left(\log x^{3}-2\right)\right)=0 \Leftrightarrow \log x\left(\log x^{3}-2\right)=1 \\
\Leftrightarrow 3 \log ^{2} x-2 \log x-1=0
\end{gathered}
$$

By substituting $\log x=t$, the last equation transforms into $3 t^{2}-2 t-1=0$, whose solutions are $t_{1}=1$ and $t_{2}=-\frac{1}{3}$. The solution $t=-\frac{1}{3}$ is rejected because from the condition we have that $t>\frac{2}{3}$. For $t=1$, we get $x=10$.

2AB. Determine the triangle with the largest area, if the lengths of its sides $a, b, c$ satisfy the condition: $1<a \leq 2 \leq b \leq 5<c \leq 6$.