# Exponential function

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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.

## Properties of exponential functions

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