Step by step
The question
Question: prove that log₂ 3 is irrational.
1 Assume the opposite
Suppose the opposite: log₂ 3 is rational, log₂ 3 = m/n, with m and n whole numbers.
log₂ 3 = mn
3 is bigger than 1, so the log is positive.
m and n are positive.
log₂ 3 > 0 (3 > 1), so m, n > 0
Careful!
The log is a power, not a number times 2.
Careful: log₂ 3 is the power of 2 that gives 3
2 Work it out
A log is a power: log₂ 3 is the power of 2 that gives 3.
2mn = 3
Raise both sides to the power n: 2ᵐ = 3ⁿ.
2m = 3n
(2^(m/n))ⁿ = 2ᵐ
3 Find the contradiction
The left side is even.
The right side is odd.
An even number can never equal an odd number!
2ᵐ is even, 3ⁿ is odd ✗
The opposite led to nonsense, so log₂ 3 is irrational.
So log₂ 3 is irrational. ∎
4 Check
A calculator: log₂ 3 is about 1.584963, the digits going on with no pattern.
log₂ 3 ≈ 1.584963...
The closest pair: 2 to the 8 is 256, 3 to the 5 is 243.
Close, but never equal.
Right!
2⁸ = 256, 3⁵ = 243: close, not equal
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.
Answer
So log₂ 3 is irrational. ∎