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

Base

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

Algebraic Properties

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

- relates Exponential Function and Trigonometric Functions
- $e^{ix} = \cos x + i \sin x$ where $i = \sqrt {-1}$

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