Exponential Functions

  • $e^x$, or $\exp x$ is the exponential function
  • Logarithm is inverse of exponent
  • exp-log.png


Base

  • "Natural" base is $e = 2.718281828...$, where $e$ is the Euler's Number
  • can be any other base: $a^x$


Algebraic Properties

  • $e^x e^y = e^{x + y}$
  • $(e^x)^y = e^{xy}$
  • $\cfrac{d}{dx} e^x = e^x$
  • $\int e^x\, dx = e^x + C$


Euler's Formula:


Taylor Expansion

Can expand $e^x = 1 + x + \cfrac{1}{2!}\, x^2 + \cfrac{1}{3!}\, x^3 + \cfrac{1}{4!}\, x^4 + \ ... \ = \sum\limits_{k=0}^{\infty} \cfrac{1}{k!} x^k$


Sources

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