# limits of natural logarithm

The parent entry (http://planetmath.org/NaturalLogarithm) defines the natural logarithm^{} as

$\mathrm{ln}x={\displaystyle {\int}_{1}^{x}}{\displaystyle \frac{1}{t}}dt\mathit{\hspace{1em}\hspace{1em}}(x>0)$ | (1) |

and derives the

$$\mathrm{ln}xy=\mathrm{ln}x+\mathrm{ln}y$$ |

which implies easily by induction^{} that

$\mathrm{ln}{a}^{n}=n\mathrm{ln}a.$ | (2) |

Basing on (1), we prove here the

Theorem. The function^{} $x\mapsto \mathrm{ln}x$ is strictly increasing and continuous^{} on ${\mathbb{R}}_{+}$. It has the limits

$\underset{x\to +\mathrm{\infty}}{lim}\mathrm{ln}x=+\mathrm{\infty}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\underset{x\to 0+}{lim}\mathrm{ln}x=-\mathrm{\infty}.$ | (3) |

*Proof.* By the above definition, $\mathrm{ln}x$ is differentiable^{}:

$$\frac{d}{dx}\mathrm{ln}x=\frac{1}{x}>\mathrm{\hspace{0.33em}0}$$ |

Accordingly, $\mathrm{ln}x$ is also continuous and strictly increasing.

Let $M$ be an arbitrary positive number. We have $\mathrm{ln}2={\int}_{1}^{2}\frac{dt}{t}>0$. There exists a positive integer $n$ such that $n\mathrm{ln}2>M$ (see Archimedean property). By (2) we thus get $\mathrm{ln}{2}^{n}>M$, and since $\mathrm{ln}x$ is strictly increasing, we see that

$$\mathrm{ln}x>M\mathit{\hspace{1em}}\forall x>{2}^{n}.$$ |

Hence the first limit assertion is true. Now $$. If $x>{2}^{n}$, then $\mathrm{ln}x>M$ and

$$ |

(substitution (http://planetmath.org/SubstitutionForIntegration) $xt:=u$). From this we can infer the second limit assertion.

Title | limits of natural logarithm |
---|---|

Canonical name | LimitsOfNaturalLogarithm |

Date of creation | 2014-12-12 10:15:50 |

Last modified on | 2014-12-12 10:15:50 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 11 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 33B10 |

Related topic | ImproperLimits |

Related topic | GrowthOfExponentialFunction |

Related topic | FundamentalTheoremOfCalculusClassicalVersion |

Related topic | DifferentiableFunctionsAreContinuous |