Solving Equations Involving Exponential and Logarithmic Functions

## Introduction

In this article, we explore the solutions of exponential equations involving variables in the exponent. By applying both algebraic and logarithmic methods, we gain insights into how to solve complex equations and understand their real and complex roots. This guide is designed to help students and enthusiasts in the field of mathematics and related sciences, including SEO optimization and content creation for digital platforms like Google.

Algebraic Solutions of Exponential Equations

Let us consider the equation: (2x^{x1^2} 2x^{3x-1^2})

Step-by-Step Solution

Step 1: Start by equating the exponents since the bases are the same. Therefore, we have:

(x1^2 3x - 1^2)

Step 2: Simplify and solve for (x1):

(x1^2 - 3x - 1^2 0)

Step 3: This is a quadratic equation in the form of:

(8x - 8x^2 0)

Step 4: Factor out common terms:

(8x(1 - x) 0)

Step 5: Solve for (x):

(x 0) or (x 1)

Step 6: Check for the validity of each solution in the original equation:

When (x 0), the left side is undefined, thus it is a rejected solution.

Therefore, the valid solution is:

(x 1)

Logarithmic Solutions of Exponential Equations

Let us consider the equation: (2x^{x1^2} 2x^{3x-1^2}) using logarithmic functions.

Step 1: Take the natural logarithm of both sides:

(ln(2x^{x1^2}) ln(2x^{3x-1^2}))

Step 2: Apply the logarithm power rule:

(x1^2 cdot ln(2x) (3x - 1)^2 cdot ln(2x))

Step 3: Simplify:

((x1^2 - (3x - 1)^2) cdot ln(2x) 0)

Step 4: Solve the equation in two parts:

(ln(2x) 0)

Step 5: Solve for (x):

(2x 1)

Therefore: (x frac{1}{2})

Step 6: The other part of the equation:

(x1^2 - (3x - 1)^2 0)

Step 7: Solve for (x):

(x1 - (3x - 1) 0)

Therefore: (x 1)

Graphical and Complex Solutions

Exploring the solutions of the equation (2x^{x1^2} 2x^{3x-1^2}) graphically:

The solutions can be visualized using a graph, which confirms that the equation has two real roots:

(x1 frac{1}{2})

(x2 1)

For complex solutions, the equation can have an infinite number of roots depending on the context. However, for practical purposes, the real roots are the most relevant.

Conclusion

By applying algebraic and logarithmic methods, we have successfully solved the given exponential equation and explored its roots. Understanding these methods is crucial for solving similar equations and for optimizing content for digital platforms like Google. The keywords for SEO optimization include 'exponential equations', 'logarithmic functions', and 'algebraic solutions', making the content more searchable and relevant.