ex Calculator

Calculate the exponential function e raised to any power x, with Taylor series terms and step-by-step solution.

Enter Exponent Value

Result

ex
7.38905610
Input x 2
e^x (exact) 7.38905610
ln(e^x) verification 2
1/e^x = e^(-x) 0.13533528
e (Euler's number) 2.71828183

Step-by-Step Solution

e^2 = 7.38905610

Understanding the Exponential Function ex

The exponential function ex is one of the most important functions in mathematics. The base e (Euler's number) is an irrational constant approximately equal to 2.71828182845904. The function ex is unique because it is its own derivative: d/dx(ex) = ex. This property makes it fundamental in calculus, differential equations, and mathematical modeling.

Key Properties of ex

Taylor Series

e^x can be expressed as an infinite series centered at x = 0.

e^x = 1 + x + x^2/2! + x^3/3! + ...

Derivative Property

The function e^x is the only function equal to its own derivative.

d/dx(e^x) = e^x

Inverse Function

The natural logarithm ln(x) is the inverse of e^x.

ln(e^x) = x, e^(ln x) = x

Product Rule

Exponentials with the same base multiply by adding exponents.

e^a * e^b = e^(a+b)

Power Rule

An exponential raised to a power multiplies the exponents.

(e^a)^b = e^(a*b)

Special Values

Key reference points for the exponential function.

e^0 = 1, e^1 = 2.71828...
e^(-inf) = 0, e^(inf) = inf

The Taylor Series Expansion

The exponential function can be represented as an infinite power series, known as the Taylor series (or Maclaurin series) expansion. This series converges for all real and complex values of x:

ex = 1 + x/1! + x2/2! + x3/3! + x4/4! + ...

Each term xn/n! becomes progressively smaller because n! (n factorial) grows much faster than xn for any fixed x. This rapid convergence makes the Taylor series a practical method for computing ex to arbitrary precision.

Euler's Number (e)

Euler's number e = 2.71828182845904... can be defined in several equivalent ways: as the limit of (1 + 1/n)n as n approaches infinity, as the sum of the series 1/0! + 1/1! + 1/2! + 1/3! + ..., or as the unique number whose natural logarithm is 1. It appears throughout mathematics, from compound interest to probability theory to fluid dynamics.

Practical Applications

  • Compound Interest: A = Pert gives the continuously compounded amount, where P is principal, r is rate, and t is time.
  • Radioactive Decay: N(t) = N0e-lambda*t models exponential decay in physics and chemistry.
  • Population Growth: P(t) = P0ekt models unbounded population growth in biology.
  • Signal Processing: ej*omega*t (Euler's formula) is the foundation of Fourier analysis and complex exponentials.
  • Probability: The normal distribution uses e-x^2/2, and the Poisson distribution uses e-lambda.
  • Differential Equations: Solutions to linear ODEs with constant coefficients are expressed in terms of ex.

Computing ex by Hand

To compute ex by hand, use the Taylor series. For small values of x, just a few terms provide good accuracy. For example, e1 = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ... = 2.71828... After about 10 terms, the result is accurate to 6 or more decimal places.