# rational Briggsian logarithms of integers

Theorem. The only positive integers, whose Briggsian logarithms^{} are rational, are the powers (http://planetmath.org/GeneralAssociativity) $1,\mathrm{\hspace{0.17em}10},\mathrm{\hspace{0.17em}100},\mathrm{\dots}$ of ten. The logarithms^{} of other positive integers are thus irrational (in fact, transcendental numbers^{}). The same concerns also the Briggsian logarithms of the positive fractional numbers.

Proof. Let $a$ be a positive integer such that

$$\mathrm{lg}a=\frac{m}{n}\in \mathbb{Q},$$ |

where $m$ and $n$ are positive integers. By the definition of logarithm, we have ${10}^{\frac{m}{n}}=a$, which is equivalent (http://planetmath.org/Equivalent3) to

$${10}^{m}={2}^{m}\cdot {5}^{m}={a}^{n}.$$ |

According to the fundamental theorem of arithmetics^{}, the integer ${a}^{n}$ must have exactly $m$ prime divisors $2$ and equally many prime divisors $5$. This is not possible otherwise than that also $a$ itself consists of a same amount of prime divisors 2 and 5, i.e. the number $a$ is an integer power of 10.

As for any rational number^{} $\frac{a}{b}$ (with $a,b\in {\mathbb{Z}}_{+}$), if one had

$$\mathrm{lg}\frac{a}{b}=\frac{m}{n}\in \mathbb{Q},$$ |

then

$${\left(\frac{a}{b}\right)}^{n}={10}^{m},$$ |

and it is apparent that the rational number $\frac{a}{b}$ has to be an integer, more accurately a power of ten. Therefore the logarithms of all fractional numbers are irrational.

Note. An analogous theorem concerns e.g. the binary logarithms ($\mathrm{lb}a$). As for the natural logarithms^{} of positive rationals ($\mathrm{ln}a$), they all are transcendental numbers except $\mathrm{ln}1=0$.

Title | rational Briggsian logarithms of integers |
---|---|

Canonical name | RationalBriggsianLogarithmsOfIntegers |

Date of creation | 2013-03-22 17:39:55 |

Last modified on | 2013-03-22 17:39:55 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 14 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 11A51 |

Related topic | Transcendental |

Related topic | RationalSineAndCosine |

Related topic | AllUnnaturalSquareRootsAreIrrational |

Related topic | BriggsianLogarithms |