# rational Briggsian logarithms of integers

Theorem.  The only positive integers, whose Briggsian logarithms are rational, are the powers (http://planetmath.org/GeneralAssociativity)  $1,\,10,\,100,\,\ldots$  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 (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 $\displaystyle\frac{a}{b}$ (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 ($\operatorname{lb}{a}$).  As for the natural logarithms of positive rationals ($\ln{a}$), they all are transcendental numbers except  $\ln 1=0$.

Title rational Briggsian logarithms of integers RationalBriggsianLogarithmsOfIntegers 2013-03-22 17:39:55 2013-03-22 17:39:55 pahio (2872) pahio (2872) 14 pahio (2872) Theorem msc 11A51 Transcendental RationalSineAndCosine AllUnnaturalSquareRootsAreIrrational BriggsianLogarithms