Question

prove that log2 is irrational​

Answer 1

Step-by-step explanation:

Theorem: log 2 is irrational

Proof:

Assuming that log 2 is a rational number. Then it can be expressed as

\frac{a}{b} with

a and

b are positive integers (Why?). Then, the equation is equivalent to

2 = 10^{\frac{a}{b}}. Raising both sides of the equation to

b, we have

2^b = 10^a. This implies that

2^b = 2^a5^a. Notice that this equation cannot hold (by the Fundamental Theorem of Arithmetic) because

2^b is an integer that is not divisible by 5 for any

b, while

2^a5^a is divisible by 5. This means that log 2 cannot be expressed as

\frac{a}{b} and is therefore irrational which is what we want to show