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Homerational Briggsian logarithms of integers

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# rational Briggsian logarithms of integers

Theorem. The only positive integers, whose Briggsian logarithms are rational, are the powers $1,\,10,\,100,\,\ldots$ 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

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

where $m$ and $n$ are positive integers. By the definition of logarithm, we have $\displaystyle 10^{{\frac{m}{n}}}=a$, which is equivalent 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 $\displaystyle\frac{a}{b}$ (with $a,\,b\in\mathbb{Z}_{+}$), if one had

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

then

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

and it is apparent that the rational number $\displaystyle\frac{a}{b}$ 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 ($\lb{a}$). As for the natural logarithms of positive rationals ($\ln{a}$), they all are transcendental numbers except $\ln 1=0$.

## Mathematics Subject Classification

11A51*no label found*

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## Comments

## Minor word detail

In the note, do you mean "analogous" rather than "analogical"?

## Re: Minor word detail

Thank you -- I have corrected it.