To solve the equation \( 2x = 2^x \), let's proceed with the following steps:

1. Recognize that the equation \( 2x = 2^x \) can be solved by inspection for simple values of \( x \).

2. Try simple integer values of \( x \):
   - For \( x = 0 \):
     \[
     2(0) = 2^0 \implies 0 = 1 \quad \text{(false)}
     \]
   - For \( x = 1 \):
     \[
     2(1) = 2^1 \implies 2 = 2 \quad \text{(true)}
     \]
   - For \( x = 2 \):
     \[
     2(2) = 2^2 \implies 4 = 4 \quad \text{(true)}
     \]
   - For \( x = 3 \):
     \[
     2(3) = 2^3 \implies 6 = 8 \quad \text{(false)}
     \]

3. Therefore, the values \( x = 1 \) and \( x = 2 \) satisfy the equation.

To confirm that these are the only solutions, we can consider the behavior of the functions \( f(x) = 2x \) and \( g(x) = 2^x \). The function \( f(x) = 2x \) is a linear function, and \( g(x) = 2^x \) is an exponential function.

- The linear function \( f(x) = 2x \) increases at a constant rate.
- The exponential function \( g(x) = 2^x \) increases at an accelerating rate.

The graphs of these functions intersect at \( x = 1 \) and \( x = 2 \). Beyond these points, the exponential function \( g(x) \) grows much faster than the linear function \( f(x) \), ensuring no further intersections.

Thus, the solutions to the equation \( 2x = 2^x \) are:

\[
x = 1 \quad \text{and} \quad x = 2
\]

Would you like more details or have any questions?

Here are 5 numbered questions you might want to ask next:
1. Can you explain how to graphically solve equations involving exponential and linear functions?
2. What are some other methods to solve exponential equations?
3. How can I solve \( 3x = 3^x \) using a similar approach?
4. Can we use logarithms to solve the equation \( 2x = 2^x \)?
5. What is the general behavior of exponential functions compared to linear functions?

**Tip:** When dealing with equations involving both linear and exponential terms, graphing can be a helpful visual aid to understand their points of intersection.

Learn how to solve the equation 2x = 2^x step-by-step with detailed explanations and graphical analysis. This problem involves understanding exponential and linear functions and their intersections. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation