Proof that log 2 is an irrational number
Before doing the proof, let us recall two things: (1) rational numbers are numbers that can be expressed as where
and
are integers, and
not equal to
; and (2) for any positive real number
, its logarithm to base
is defined to be a number
such that
. In proving the statement, we use proof by contradiction.
Theorem: log 2 is irrational
Proof:
Assuming that log 2 is a rational number. Then it can be expressed as with
and
are positive integers (Why?). Then, the equation is equivalent to
. Raising both sides of the equation to
, we have
. This implies that
. Notice that this equation cannot hold (by the Fundamental Theorem of Arithmetic) because
is an integer that is not divisible by 5 for any
, while
is divisible by 5. This means that log 2 cannot be expressed as
and is therefore irrational which is what we want to show.