rational Briggsian logarithms of integers
Theorem. The only positive integers, whose Briggsian logarithms are rational, are the powers (http://planetmath.org/GeneralAssociativity) 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 be a positive integer such that
where and are positive integers. By the definition of logarithm, we have , which is equivalent (http://planetmath.org/Equivalent3) to
According to the fundamental theorem of arithmetics, the integer must have exactly prime divisors and equally many prime divisors . This is not possible otherwise than that also itself consists of a same amount of prime divisors 2 and 5, i.e. the number is an integer power of 10.
As for any rational number (with ), if one had
and it is apparent that the rational number has to be an integer, more accurately a power of ten. Therefore the logarithms of all fractional numbers are irrational.
|Title||rational Briggsian logarithms of integers|
|Date of creation||2013-03-22 17:39:55|
|Last modified on||2013-03-22 17:39:55|
|Last modified by||pahio (2872)|