Abstract
Series of exponential equations in the form of were solved graphically, numerically and analytically. The analytical solution was derived in terms of Lambert-W function. A general numerical solution for any y is found in terms of n or in base y. A solution is close to the fine structure constant. The equation which provided the solution as the fine structure constant was derived in terms of the fundamental constants.
Highlights
Exponential equations are widely used in natural and social sciences
The analytical solution was derived in terms of Lambert-W function
We considered series of exponential equations and solved them graphically, numerically, and analytically in terms of Lambert-W function
Summary
Exponential equations are widely used in natural and social sciences. We considered series of exponential equations and solved them graphically, numerically, and analytically in terms of Lambert-W function. One equation connected to the fine structure constant, was derived in terms of the fundamental constants and led to a new equation. The Lambert-W function for real variables is defined by the equation W ( x) exp W ( x) = x [1] [2] [3] [4] and it has applications in Planks spectral distribution law [5] [6], QCD renormalization [7], solar cells [8], bio-chemical kinetics [9], optics [10], population growth. Considering the series of exponential equations defined by the following equation x (1.1). We are focusing on the non-trivial solutions
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