Step by step
The question
Question: prove that log₃ 5 is irrational.
1 Assume the opposite
Suppose the opposite: log₃ 5 is rational, log₃ 5 = m/n, with m and n whole numbers.
5 is bigger than 1, so the log is positive.
m and n are positive.
Careful!
The log is a power, not a number times 3.
2 Work it out
A log is a power: log₃ 5 is the power of 3 that gives 5.
Raise both sides to the power n: 3ᵐ = 5ⁿ.
3 Find the contradiction
Prime factors: 3 to the m is only 3s, 5 to the n is only 5s.
A number has only one set of prime factors.
They can never be equal!
The opposite led to nonsense, so log₃ 5 is irrational.
4 Check
A calculator: log₃ 5 is about 1.464974, the digits going on with no pattern.
The closest pair: 3 to the 3 is 27, 5 to the 2 is 25.
Close, but never equal.
Right!
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.