To solve the equation \((-5)^x = 5\), let's first analyze the properties of exponentiation and the base.

### Step-by-Step Solution:
1. **Consider the nature of the base:**
   - For the equation \((-5)^x = 5\), the base \(-5\) is a negative number.
   - A negative base raised to a power only results in a positive value when the exponent \( x \) is an even integer. For odd integer exponents, the result would be negative.

2. **Check if \( x \) can be a real number:**
   - If \( x \) were an even integer, \((-5)^x\) would yield a positive value, but not necessarily equal to \( 5 \).
   - If \( x \) were an odd integer, \((-5)^x\) would yield a negative value, which can never be equal to \( 5 \).

3. **Conclusion for real solutions:**
   - There is no real number \( x \) for which \((-5)^x = 5\).

4. **Complex solution:**
   - If we consider complex numbers, we use the fact that any real number \( a \) can be written in exponential form: 
     \[
     (-5)^x = e^{x \ln(-5)}
     \]
   - Since \(\ln(-5)\) is undefined in the real numbers, we need to express \(-5\) as \( 5e^{i\pi} \):
     \[
     (-5)^x = \left( 5e^{i\pi} \right)^x = 5^x \cdot e^{ix\pi}
     \]
   - Setting this equal to 5:
     \[
     5^x \cdot e^{ix\pi} = 5
     \]
   - This simplifies to:
     \[
     5^x \cdot e^{ix\pi} = 5^1 \cdot e^{i \cdot 0}
     \]
     Thus, equating magnitudes and angles, we get:
     \[
     x = 1 \quad \text{and} \quad x\pi = 2k\pi \quad \text{for integer } k
     \]
     \[
     x\pi = 0 \quad \Rightarrow \quad x = 1
     \]

   - Therefore, the solution for \( x \) in the complex plane would be:
     \[
     x = 1 + \frac{2k}{\pi} i, \quad k \in \mathbb{Z}
     \]

### Final Answer:
For **real numbers**, there is no solution. For **complex numbers**, the solution is:
\[ x = 1 + \frac{2k}{\pi} i, \quad k \in \mathbb{Z} \]

Would you like more details on any of the steps, or have any other questions?

Here are 5 related questions:
1. How do you solve exponential equations with a negative base?
2. What are the properties of exponentiation in the complex plane?
3. How can you convert negative numbers into their exponential form?
4. What does it mean for an exponentiation equation to have no real solutions?
5. How do we derive the complex logarithm of a negative number?

**Tip:** When dealing with negative bases in exponentiation, always consider both real and complex solutions separately.

Explore the step-by-step solution to the equation (-5)^x = 5, analyzing real and complex solutions. This problem covers concepts like exponentiation of negative numbers and the use of logarithms in complex number theory. Suitable for university-level mathematics students.

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