To solve the equation \((x-1)^x = 1024\), follow these steps:

### Step 1: Recognize possible powers of 1024
We start by recognizing that \(1024\) is a power of 2. Specifically:
\[
1024 = 2^{10}
\]

### Step 2: Set up the equation
Given that \((x-1)^x = 2^{10}\), the equation becomes:
\[
(x-1)^x = 2^{10}
\]

### Step 3: Consider \(x-1\) as a power of 2
To simplify, let's assume \(x-1 = 2^a\) for some integer \(a\). Then the equation becomes:
\[
(2^a)^x = 2^{10}
\]
Which simplifies to:
\[
2^{ax} = 2^{10}
\]

### Step 4: Equate exponents
Since the bases are the same, equate the exponents:
\[
ax = 10
\]

### Step 5: Substitute back \(x-1\)
Recall that \(x-1 = 2^a\), so:
\[
x = 2^a + 1
\]
Substituting into \(ax = 10\):
\[
a(2^a + 1) = 10
\]

### Step 6: Test possible integer values for \(a\)
Let's test integer values of \(a\) that satisfy \(a(2^a + 1) = 10\):

- For \(a = 1\): \(1(2^1 + 1) = 1 \times 3 = 3\)
- For \(a = 2\): \(2(2^2 + 1) = 2 \times 5 = 10\)

So, \(a = 2\) works. Substitute \(a = 2\) back:
\[
x = 2^2 + 1 = 4 + 1 = 5
\]

### Conclusion
The solution to the equation \((x-1)^x = 1024\) is:
\[
x = 5
\]

Would you like any more details or have any questions?

---

Here are 5 related questions to explore further:
1. What is the general method for solving exponential equations?
2. Can this method be applied to solve other similar equations like \((x-2)^x = 512\)?
3. How do we determine the powers of numbers efficiently?
4. What happens when the base of the exponent is not an integer?
5. Can you derive the properties of exponents from first principles?

**Tip:** When solving exponential equations, always try to express the equation in terms of the same base if possible.

Learn how to solve the exponential equation (x-1)^x = 1024 step-by-step. This problem involves recognizing powers of numbers, setting up and solving exponential equations, and testing integer solutions. Suitable for high school students and those interested in advanced algebraic problem-solving.

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