linguistic:voice_change
Logarithms

Perhaps the numbers most easy to prove irrational are certain logarithms. Here is a proof by contradiction that log 3 is irrational (log 3 ≈ 1.58 > 0).

Assume log 3 is rational. For some positive integers m and n, we have

It follows that

The number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log 3 is rational (and so expressible as a quotient of integers m / n with n ≠ 0). The contradiction means that this assumption must be false, i.e. log 3 is irrational, and can never be expressed as a quotient of integers m / n with n ≠ 0.

Cases such as log 2 can be treated similarly.