exponential

Exponential function

For base $a>0$: $f(x)=a^x=e^{x\ln a}$, defined for all $x\in \mathbb{R}$.

The exponential function is continuous on $\mathbb{R}$ and strictly increasing for $a>1$, decreasing for $0<a<1$, constant for $a=1$.

The base must be positive: $a^x$ for $a<0$ is not defined for general real exponents (e.g. $(-2{)}^{1/2}=\sqrt{-2}\notin \mathbb{R}$).

Its inverse is the logarithm: ${\log }_a(a^x)=x$.

The exponential turns addition into multiplication

$$a^{x+y}=a^x\cdot a^y$$ $$\exp \sum a_i=\prod \exp a_i$$ $$\log \prod a_i=\sum \log a_i$$

Simplify $\exp \sum a_i$

$$\prod \exp a_i$$

Geometric growth / decay / sequence $a_n=a_0{r}^n$ for $n=0,1,2,\dots$ is another term for exponential growth / decay but with discrete time steps.

A geometric progression

euler’s number

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