Exponential function to base e

Calculator and formula for calculating the power value of base e

Exponential function calculator (base e)

What is calculated?

This function computes the power value of the given exponent for base e. The argument must be a real number. The Exp function for complex numbers can be found here.

Input values


Result
The result is shown with the selected number of decimal places

Exponential function info

Properties

Exponential function base e:

  • Natural exponential function
  • Base e ≈ 2.71828...
  • Variable is in the exponent
  • Strictly increasing
  • Range: (0, ∞)

Note: The natural exponential function is the inverse of the natural logarithm.

Examples
e⁰ = 1
Any number to the power of 0 equals 1
e¹ ≈ 2.71828
Euler's number
e² ≈ 7.38906
Square of e
e³ ≈ 20.08554
Cube of e

Formula of the exponential function (base e)

General form
\[f(x) = e^x\] Natural exponential function
Power series
\[e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}\] Series expansion
Expanded series
\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\] Expanded form
Rewrite to base 10
\[e^x = 10^{x \cdot \log_{10}(e)}\] Using base-10 logarithm
Derivative
\[\frac{d}{dx}e^x = e^x\] Special property
Antiderivative
\[\int e^x dx = e^x + C\] Integral of e^x

Calculation example

Example: calculate e⁴
\[e^4 = 54.59815...\]

Euler's number e ≈ 2.71828 is used as base with exponent 4.

Stepwise calculation with power series:
\[e^4 = 1 + 4 + \frac{4^2}{2!} + \frac{4^3}{3!} + \frac{4^4}{4!} + \cdots\] \[= 1 + 4 + \frac{16}{2} + \frac{64}{6} + \frac{256}{24} + \cdots\] \[= 1 + 4 + 8 + 10.667 + 10.667 + \cdots\] \[≈ 54.59815\]

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