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The exponential function is a function of the form $ f(x)=e^x $. Euler's number, $ e $, is the base of the natural logarithm. Sometimes, in general, the term exponential function can refer to functions of the form $ f(x)=ka^x $, where $ a,k $ are constants. The variable $ a $ here is the base. The function is sometimes denoted $ e^x=\exp(x) $ when superscripts are inappropriate.

Properties of exponential functions

  • $ e^{x}e^{y} = e^{x+y} $
  • $ a^0=1 $
  • $ \lim_{x\to\infty}a^x=0\quad\text{if }0\leq a<1 $
  • $ \lim_{x\to\infty}a^x=\infty\quad\text{if }a>1 $
  • $ \frac{d}{dx}\big(a^x\big)=a^x\ln(a) $
  • $ \frac{d}{dx}\big(e^x\big)=e^x $
  • $ \int a^xdx=\dfrac{a^x}{\ln(a)}+C $
  • $ \int e^xdx= e^x+C $
  • $ \int\limits_{-\infty}^0e^xdx=1 $
  • $ \int\limits_0^1e^xdx=e-1 $

Definitions

1.The limit definition:

$ e^x = \lim_{n \to \infty} \left(1+\dfrac{x}{n}\right)^n $

2.The infinite series:

$ e^x = \sum_{n=0}^{\infty} \dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2!} + \cdots $

3.The inverse of the natural logarithm, where $ e^x $ is the number $ y>0 $ such that

$ \int_{1}^{y} \frac{1}{t}dt = x $

4.The unique solution to the initial value problem

$ y'=y, y(0)=1 $


See also

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