The equation \( 3^x = -50 \) does not have a solution in the real numbers because an exponential function with a positive base (in this case, \( 3 \)) always produces positive outputs for any real value of \( x \). Therefore, there is no real \( x \) such that \( 3^x = -50 \).

However, if you are considering solutions in the complex number domain, we can proceed using logarithms and Euler's formula to find the complex solutions. Here's how to solve it in the complex plane:

1. Start with \( 3^x = -50 \).
2. Rewrite the equation as \( 3^x = 50e^{i\pi} \) since \( -50 \) can be written in polar form as \( 50e^{i\pi} \).
3. Take the natural logarithm of both sides:
   \[
   x \ln(3) = \ln(50e^{i\pi}) = \ln(50) + i\pi
   \]
4. Solve for \( x \):
   \[
   x = \frac{\ln(50) + i\pi}{\ln(3)}
   \]

This gives the general complex solution. Additionally, since \( e^{i(\pi + 2k\pi)} = e^{i\pi} \) for any integer \( k \), the full solution includes multiple values for \( x \) based on \( 2k\pi \):

\[
x = \frac{\ln(50) + i(\pi + 2k\pi)}{\ln(3)} \quad \text{for any integer} \, k.
\]

Let me know if you need a more detailed explanation of any step!

---
Would you like more details on solving in the complex plane or have any other questions? Here are 5 related questions to explore:

1. How do you convert numbers to polar form?
2. What is Euler's formula and how is it used in complex exponentials?
3. Can negative numbers be solutions to exponential equations in real numbers?
4. How do natural logarithms work with complex numbers?
5. What are principal values when solving complex equations?

**Tip**: When solving equations involving complex numbers, always check if the problem asks for real or complex solutions.

This page provides a step-by-step solution to the exponential equation 3^x = -50, showing why there are no real solutions and how to find complex solutions using logarithms and Euler's formula. Key concepts include exponential functions, complex numbers, and logarithmic identities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation