2. In the product $5^{-2} \cdot 5^{-4} \cdot 5^{-8} \cdot \ldots \cdot 5^{-x}$, where the exponents form a geometric sequence, determine $x$ such that $5^{-2} \cdot 5^{-4} \cdot 5^{-8} \cdot \ldots \cdot 5^{-x}=5^{-16382}$.
2. We write the sum of the terms of the geometric sequence $-2+(-4)+(-8)+\ldots+(-x)=-16382$, from which we read the first term, the common ratio, and $x=a_{n}$. We use the formula for the sum of the first terms of a geometric sequence $s_{n}=a_{1} \cdot \frac{k^{n}-1}{k-1}$. We substitute the data and calculate that $n=13$. We use the relationship that $-x=a_{13}=a_{1} \cdot k^{12}=-8192$.

Conclusion $n=-2-4-8-\ldots-x=-16382$ 1 point
Calculation of the common ratio $k=2$ and determination of the first term $a_{1}=-2$, identification of $x=a_{n}$ and $s_{n}=-16382$
$1+1$ point
Application of the formula $-16382=-2 \cdot \frac{2^{n}-1}{2-1}$ ..... 1 point
Solution $n=13$. ..... 1 point
Calculation $x=8192$ ..... 1 point