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