Solving the Exponential Equation e^{3x} 6e^{4-x^2}

This article provides a detailed step-by-step solution to the given exponential equation, e^{3x} 6e^{4-x^2}, using properties of logarithms and algebraic manipulation. Understanding these methods is crucial for solving a variety of related equations in both academic and real-world contexts.

Introduction

An exponential equation often appears in various fields, including physics, finance, and engineering. The equation e^{3x} 6e^{4-x^2} is a specific example where we need to find the value of x. This article will explore the process of solving such an equation, focusing on the application of logarithmic properties and algebraic techniques.

Understanding the Equation

The given equation e^{3x} 6e^{4-x^2} consists of two exponential terms. To solve this equation, we can use the property of exponents that states (frac{a^m}{a^n} a^{m-n}), and the property of logarithms, which allows us to solve for the variable by isolating the exponent.

Solution Steps

Let's break down the solution into manageable steps:

Start with the given equation: (e^{3x} 6e^{4-x^2}). Isolate the exponential term on one side: [e^{3x} frac{6e^4}{e^{x^2}}] Multiply both sides by (e^{x^2}) to get (e^{3x} cdot e^{x^2} 6e^4). Combine the exponents on the left side: [e^{x^2 3x} 6e^4] Take the natural logarithm (ln) of both sides to simplify the equation: [ln(e^{x^2 3x}) ln(6e^4)] Use the property of logarithms that (ln(e^a) a) to simplify the left side and the property of logarithms (ln(ab) ln(a) ln(b)) for the right side: [x^2 3x ln(6e^4) ln(6) ln(e^4)] Simplify the right side: [x^2 3x ln(6) 4] Complete the square to solve for (x). First, subtract 4 from both sides: [x^2 3x - 4 ln(6)] Add (left(frac{3}{2}right)^2 frac{9}{4}) to both sides: [x^2 3x frac{9}{4} ln(6) frac{9}{4}] Factor the left side: [left(x frac{3}{2}right)^2 ln(6) frac{9}{4}] Take the square root of both sides: [x frac{3}{2} pm sqrt{ln(6) frac{9}{4}}] Solve for (x): [x -frac{3}{2} pm sqrt{ln(6) frac{9}{4}}] Further simplify the term under the square root: [x -frac{3}{2} pm sqrt{ln(6) frac{9}{4}} -frac{3}{2} pm sqrt{frac{4ln(6) 9}{4}}] The solution can be written as: [x -frac{3}{2} pm frac{sqrt{4ln(6) 9}}{2}]

Conclusion

By systematically applying several mathematical properties, including the properties of exponents and logarithms, we can solve the given equation e^{3x} 6e^{4-x^2}). The final solution involves using the square root of a logarithmic term, demonstrating the importance of understanding these fundamental mathematical concepts.

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