Step by step
The question
Question: prove that log₃ 2 is irrational.
1 Assume the opposite
Suppose the opposite: log₃ 2 is rational, log₃ 2 = m/n, with m and n whole numbers.
log₃ 2 = mn
2 is bigger than 1, so the log is positive.
m and n are positive.
log₃ 2 > 0 (2 > 1), so m, n > 0
Careful!
The log is a power, not a number times 3.
Careful: log₃ 2 is the power of 3 that gives 2
2 Work it out
A log is a power: log₃ 2 is the power of 3 that gives 2.
3mn = 2
Raise both sides to the power n: 3ᵐ = 2ⁿ.
3m = 2n
(3^(m/n))ⁿ = 3ᵐ
3 Find the contradiction
The left side is odd.
The right side is even.
An even number can never equal an odd number!
2ⁿ is even, 3ᵐ is odd ✗
The opposite led to nonsense, so log₃ 2 is irrational.
So log₃ 2 is irrational. ∎
4 Check
A calculator: log₃ 2 is about 0.630930, the digits going on with no pattern.
log₃ 2 ≈ 0.630930...
The closest pair: 3 to the 5 is 243, 2 to the 8 is 256.
Close, but never equal.
Right!
3⁵ = 243, 2⁸ = 256: close, not equal
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.
Answer
So log₃ 2 is irrational. ∎