Math Problem Statement

Déterminer l'ensemble des solutions dans ℝ de: e^(-2x - 5) = e^x

Solution

The problem you're working with asks to solve the equation:

e2x5=exe^{-2x - 5} = e^x

in the real numbers R\mathbb{R}, and the answer should be given in the form of a set of solutions.

Solution:

Since the equation involves exponential functions with the same base ee, we can equate the exponents directly:

2x5=x-2x - 5 = x

Now, let's solve for xx:

  1. Move all terms involving xx to one side: 2xx=5-2x - x = 5 3x=5-3x = 5

  2. Solve for xx by dividing both sides by 3-3: x=53=53x = \frac{5}{-3} = -\frac{5}{3}

Final Answer:

The solution set is:

{53}\left\{-\frac{5}{3}\right\}

Would you like more details or have any questions?


Here are 5 related questions for further exploration:

  1. How do we solve exponential equations with different bases?
  2. What happens if we add logarithms to both sides of an exponential equation?
  3. Can exponential equations always be solved algebraically?
  4. How would the solution change if the equation was e2x5=e2xe^{-2x - 5} = e^{2x}?
  5. How does the exponential function behave as xx approaches infinity?

Tip: When solving exponential equations, it’s helpful to simplify by isolating the exponents or converting to logarithmic form when needed.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponential Equations

Formulas

e^a = e^b implies a = b

Theorems

Properties of exponential functions

Suitable Grade Level

Grades 9-12