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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

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