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The given equation is \( a^b = x^b \), and the question asks for when \( a = b \), based on the conditions given for \( x \).

We are looking for the values of \( x \) under which \( a = b \) holds. Let's analyze each option carefully:

### Step 1: Analyzing the equation
The equation is \( a^b = x^b \), and it simplifies to:
\[
a = x
\]
If \( b \neq 0 \), raising both sides of the equation \( a^b = x^b \) to the power of \( \frac{1}{b} \), implies that \( a = x \), unless some restrictions or specific values of \( x \) and \( b \) change this behavior.

So, we are essentially looking for cases where \( a = b \) under specific values of \( x \).

### Step 2: Checking the options
#### Option (a) \( x > 0 \) and \( x = 1 \)
If \( x = 1 \), then \( a = 1 \), and the equation becomes \( a = b \), because any power of 1 remains 1. Therefore, \( b = a = 1 \), which satisfies the condition. Hence, option (a) is **true**.

#### Option (b) \( x > 0 \) and \( x \neq 1 \)
If \( x \neq 1 \), then \( a^b = x^b \) does not guarantee \( a = b \) unless \( a \) and \( x \) are equal, which contradicts the condition \( x \neq 1 \). Hence, option (b) is **false**.

#### Option (c) \( x < 0 \) and \( x \neq 1 \)
If \( x \) is negative and \( x \neq 1 \), this would lead to issues with taking powers of negative numbers (depending on whether \( b \) is an integer or not). Even in cases where powers are defined, \( a = b \) would not hold. Hence, option (c) is **false**.

#### Option (d) \( x < 0 \) and \( x = 1 \)
This option is contradictory because it says \( x < 0 \) but also \( x = 1 \). This is impossible. Hence, option (d) is **false**.

### Conclusion:
The correct answer is **(a)**, where \( x > 0 \) and \( x = 1 \).

Would you like further details or have any questions about this? Here are some related questions you might consider:
1. How would the solution change if \( b \) were zero?
2. What if \( a^b = x^c \) instead of \( a^b = x^b \)?
3. How does the solution change for non-integer exponents?
4. What would happen if \( x \) were a complex number?
5. How would the equation behave if negative powers were involved?

**Tip**: When dealing with exponentials and variable bases, always consider special cases like \( x = 1 \) or \( x = 0 \), as they often simplify the equation greatly.

If a^b = x^b, a = b when:
a) x > 0 and x = 1
b) x > 0 and x ≠ 1
c) x < 0 and x ≠ 1
d) x < 0 and x = 1

Explore how to solve the equation a^b = x^b, where a = b under different conditions on x. This problem covers exponentiation and algebraic principles. Learn which condition holds true: x > 0 and x = 1. Suitable for Grades 9-12.

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