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 $\displaystyle10^{\frac{m}{n}} = a$ , 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 $\ln1 = 0$ .