You are solving the equation:

\[
6 \cdot e^{2x} = 4x
\]

Here's how to approach this:

### Step 1: Isolate the exponential term

To make the problem more manageable, we can rewrite the equation as:

\[
e^{2x} = \frac{4x}{6}
\]

which simplifies to:

\[
e^{2x} = \frac{2x}{3}
\]

### Step 2: Analyze the equation
This equation contains both an exponential and a linear term involving \(x\). Solving such an equation analytically is not straightforward due to the mixed nature of the terms.

### Step 3: Use numerical methods
Since this equation cannot be solved algebraically, it is best to solve it using a numerical approach, such as Newton's method, or graphically to find approximate solutions.

Would you like to proceed with a numerical solution or graph the functions to find approximate solutions?

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Feel free to ask about these related topics:
1. How to solve exponential equations using numerical methods?
2. What is Newton's method for solving non-linear equations?
3. How to graph exponential and linear functions?
4. What are the properties of exponential functions?
5. How to solve similar equations involving exponentials?

**Tip:** For equations with mixed exponentials and polynomials, numerical solutions or graphing tools are often more practical than algebraic methods.

Learn how to solve the exponential equation 6 · e^{2x} = 4x with a detailed step-by-step solution. This problem involves key concepts in exponential functions and numerical methods, including Newton's Method and properties of exponential functions. Suitable for students in Grades 10-12.

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