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Exponential

Tool to calculate the values of the exponential function exp(x) e(x) e^x and solve the calculations related to the function or the constant e=2.71818…

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Exponential -

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# Exponential

## Exponentiation Calculator a^b (Exponential of base a)

### What is the exponential function? (Definition)

The definition of the exponential function is the solution of the equation $f' = f$ with $f(0) = 1$, i.e. the function which is its own derivative and which has the value 1 at 0.

The exponential function is denoted by exp that is, by default, based on the number $e \approx 2.71828\ldots$ (check also the decimals of the number e).

Example: $\exp(7) = e^7 \approx 1096.633$ The $e^x$ notation is sometimes ambiguous, because $e$ may be used as a variable, prefer using the $\exp(x)$ notation.

### What are the properties of the exponential function?

The exponential has several remarkable properties

$\exp(0) = 1 \\ \exp(1) = e \approx 2.71828\ldots \\ e^(x+y) = e^x \times e^y \\ (e^x)^b = e^{bx} \\ \ln(\exp(x)) = x \\ \exp(\ln(x)) = x $$The derivative of the exponential function is the exponential function itself$$ f(x) = \exp(x) \iff f'(x) = \exp(x) $$The exponential is related to the exponentiation by the formula:$$ a^b = e^{b\ln(a)} $$In the complex plane, the exponential has several other properties (complex exponential form):$$ \exp(i x) = \cos x + i \sin x \\ \exp(a + i b) = \exp(a) ( \cos b + i \sin b ) $$The exponential function can be defined as a series expansion based on factorial and exponentiation:$$ \exp(x)=\sum _{{n=0}}^{{\infty }}{x^{n} \over n!}$\$

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