We are given the equation:

\[
|x|! = 2^{1024}
\]

This equation involves a factorial on one side and an exponential power of 2 on the other. Let's break this down step by step.

1. \( |x| \) represents the absolute value of \( x \), which means \( x \) can be positive or negative, but the factorial applies to a positive integer.
   
2. \( n! \) (the factorial) represents the product of all positive integers up to \( n \). For large values of \( n \), factorials grow extremely fast.

### Step 1: Approximate \( 2^{1024} \)
The number \( 2^{1024} \) is very large. To understand its magnitude, we can approximate it using logarithms:

\[
2^{1024} \approx 1.797 \times 10^{308}
\]

This number is astronomically large, and we'll need to find \( n \) such that:

\[
n! \approx 1.797 \times 10^{308}
\]

### Step 2: Approximate \( n! \)
Stirling's approximation is a useful tool to estimate large factorials. It is given by:

\[
n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n
\]

Taking the logarithm of both sides to compare it to \( 2^{1024} \), we have:

\[
\ln(n!) \approx n \ln(n) - n
\]

We also know that \( \ln(2^{1024}) = 1024 \ln(2) \approx 710.7 \).

Thus, we need to solve:

\[
n \ln(n) - n \approx 710.7
\]

This equation requires numerical methods or estimation to find the value of \( n \).

### Step 3: Estimate \( |x| \)
Using numerical approaches, we find that:

\[
n \approx 171!
\]

Thus, \( |x| \) must be approximately 171. Therefore:

\[
|x| \approx 171
\]

### Final Answer:
The solution to \( |x|! = 2^{1024} \) is approximately:

\[
|x| = 171
\]

### Do you have any questions or would you like further details?

Here are 5 related questions to deepen understanding:
1. How does Stirling's approximation work for large factorials?
2. Can factorials be computed efficiently for very large numbers?
3. Why does \( 2^{1024} \) result in such a large number?
4. What are some practical applications of factorials in combinatorics?
5. What are the limitations of using numerical methods to solve large equations?

**Tip:** Factorials grow faster than exponential functions; estimating large factorials often requires approximation techniques like Stirling's formula.

Explore how to solve the equation |x|! = 2^1024, utilizing Stirling's approximation and logarithms to estimate large factorials. This problem covers key mathematical concepts such as exponential growth and numerical approximation. Suitable for undergraduate students.

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