Solving the Exponential Equation 2x 3x 5x: Methods and Analysis

The equation 2x 3x 5x is a fascinating problem in the realm of mathematical analysis. Let's explore various methods to solve this equation and determine the nature of its roots.

Standard Solution

It is clear that x 1 is a solution to the equation. To verify, we can substitute x 1 into the equation:

x 1

21 31 51 rArr; 2 3 5

This confirms that x 1 is indeed a solution. Graphically, we can represent the function y 2x 3x and plot it against y 5x to visualize the intersection.

Function Analysis for x 1

Define the function f(x) 2x 3x - 5x.

Formulation of f(x)

f(x) 2/5x 3/5x - 1. Taking the derivative, we get:

Derivative of f(x)

f'(x) (2/5x log (2/5)) (3/5x log (3/5))

Since log(2/5) and log(3/5) are both negative, and 2/5x is always positive, f'(x) is negative.

Conclusion on Roots

Since f(x) is strictly decreasing, it can only cross the x-axis once, meaning there is only one real root. This root is known to be x 1. Any complex roots, if they exist, do not lie on the real number line.

Complex Solutions Using Computational Methods

For a deeper dive, let's consider the equation in a more complex context using computational methods. The function f(z) 2z 3z - 5z.

General Form of Roots

The general solution using the GRP-N method yields:

x 2Iπk/Log[2/5] Log[u]/Log[2/5]…k in Z … Real roots: x0 1.00000 Roots: Infinity x1/2 -1.04668993583378683327906044-/7.24362253938625532529586148687 I x3/4 0.821622925760757071183645051-/13.0854566768203354428049939170 I x1.000.000 0.9680264170452295325-6.8571964562400040678745604010^6 I

The method found a complex root at order 1.000.000 with computation time under 1/10 second.

Real-Valued Function Analysis

For the real-valued function f(x) 2x 3x - 5x, we analyze its first derivative:

First Derivative

f'(x) (2x log 2) (3x log 3) - 5x log 5

f'(x) 0 for x 0.1614827263138109

f'(x) 0 for all x ge; 0.1614827263138109

f'(x) 0 for all x 0.1614827263138109

The function is strictly increasing for x 0.1614827263138109 and strictly decreasing for x 0.1614827263138109. Since f'(x) is not zero for any x except at the maximum point, there is only one real root.

Conclusion

The equation 2x 3x 5x has a unique real root at x 1. The graph of the function y 2x 3x intersects y 5x at this point. Any complex roots, if they exist, are not on the real number line.