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

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

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 ${\displaystyle \frac{a}{b}}$ (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 ($\mathrm{lb}a$).  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
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