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...

Exponential - dCode

Tag(s) : Functions

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

⮞ Go to: Exponentiation (Power)

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.

The exponential has several remarkable properties

$ \exp(0) = 1 \\ \exp(1) = e \approx 2.71828\ldots \\ e^x + e^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:

$$ \exp(x)=\sum _{{n=0}}^{{\infty }}{x^{n} \over n!} $$

dCode retains ownership of the online 'Exponential' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Exponential' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Exponential' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Exponential' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.

Please, check our dCode Discord community for help requests!

NB: for encrypted messages, test our automatic cipher identifier!

exponential,function,exp,derivative

Source : https://www.dcode.fr/exponential

© 2021 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲
Thanks to your feedback and relevant comments, dCode has developed the best 'Exponential' tool, so feel free to write! Thank you!