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.