rational Briggsian logarithms of integers

Theorem.  The only positive integers, whose Briggsian logarithmsMathworldPlanetmath are rational, are the powers (http://planetmath.org/GeneralAssociativity)  1, 10, 100,  of ten.  The logarithmsMathworldPlanetmath of other positive integers are thus irrational (in fact, transcendental numbersMathworldPlanetmath).  The same concerns also the Briggsian logarithms of the positive fractional numbers.

Proof.  Let a be a positive integer such that


where m and n are positive integers.  By the definition of logarithm, we have  10mn=a,  which is equivalent (http://planetmath.org/Equivalent3) to


According to the fundamental theorem of arithmeticsMathworldPlanetmath, the integer an 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 numberPlanetmathPlanetmath ab (with  a,b+), if one had




and it is apparent that the rational number ab 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 (lba).  As for the natural logarithmsMathworldPlanetmath of positive rationals (lna), they all are transcendental numbers except  ln1=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