Exponential function

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

• $e^{x}e^{y} = e^{x+y}$

• $a^0 =1$

• $\lim_{x\to\infty}a^x=0\quad\text{if }0\leq a<1$

• $\lim_{x\to\infty}a^x=\infty\quad\text{if }a>1$

• $\frac{d}{dx}\big(a^x\big)=a^x\ln(a)$

• $\frac{d}{dx}\big(e^x\big)=e^x$

• $\int a^xdx=\dfrac{a^x}{\ln(a)}+C$

• $\int e^x dx = e^x + C$

• $\int\limits_{-\infty}^0e^xdx=1$

• $\int\limits_0^1e^xdx=e-1$

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

• Logarithmic function