what is the natural exponential function

Exponential Growth with Base e

This entry details a specific exponential mathematical function that plays a critical role in calculus, differential equations, and numerous applications across science, engineering, and finance.

Definition and Properties

The function is typically written as f(x) = ex, where e is Euler's number, an irrational constant approximately equal to 2.71828. It exhibits the fundamental property of exponential functions: that the rate of change of the function is proportional to its current value. Specifically, its derivative is itself: d/dx (ex) = ex.

Euler's Number (e)

Euler's number is defined as the limit of (1 + 1/n)n as n approaches infinity. It also arises as the sum of the infinite series Σ (1/n!), where n ranges from 0 to infinity. This constant is fundamental in mathematics and appears in various formulas, including those related to compound interest and probability.

Taylor Series Representation

The function possesses a convergent Taylor series expansion around x = 0: ex = 1 + x + x2/2! + x3/3! + ... = Σ (xn/n!), where n ranges from 0 to infinity. This representation is crucial for approximating values of the function and for analytical manipulations.

Graphical Representation

The graph of the function is strictly increasing and concave up for all real values of x. It passes through the point (0, 1), has a horizontal asymptote at y = 0 as x approaches negative infinity, and increases without bound as x approaches positive infinity.

Applications

  • Compound Interest: Models continuous compounding of interest, where the growth rate is proportional to the principal.
  • Radioactive Decay: Describes the exponential decrease in the amount of a radioactive substance over time.
  • Population Growth: Can approximate population growth in ideal conditions, where resources are unlimited.
  • Differential Equations: Appears as a solution to many linear, homogeneous differential equations with constant coefficients.
  • Probability and Statistics: Related to the normal distribution and other probability distributions.

Relationship to the Natural Logarithm

The function and the natural logarithm, denoted as ln(x) or loge(x), are inverse functions. This means that ln(ex) = x for all real numbers x, and eln(x) = x for all positive real numbers x. The natural logarithm is the logarithm to the base e.