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  $\ln 1=0$.