# natural log base

The *natural log base*, or $e$, has value

$$2.718281828459045\mathrm{\dots}$$ |

$e$ was extensively studied by Euler in the 1720’s, but it was originally discovered by John Napier.

$e$ is defined by

$$\underset{n\to \mathrm{\infty}}{lim}{\left(1+\frac{1}{n}\right)}^{n}$$ |

It is more effectively calculated, however, by using the Taylor series^{} for $f(x)={e}^{x}$ at $x=1$ to get the representation

$$e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\mathrm{\cdots}$$ |

Title | natural log base |

Canonical name | NaturalLogBase |

Date of creation | 2013-03-22 11:55:56 |

Last modified on | 2013-03-22 11:55:56 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 11 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 33B99 |

Synonym | Euler number^{} |

Synonym | Eulerian number^{} |

Synonym | Napier’s constant |

Synonym | e |

Related topic | ExampleOfTaylorPolynomialsForTheExponentialFunction |

Related topic | EIsTranscendental |

Related topic | EIsIrrationalProof |

Related topic | ApplicationOfCauchyCriterionForConvergence |