Step by step
The question
Question: prove that log₅ 7 is irrational.
1 Assume the opposite
Suppose the opposite: log₅ 7 is rational, log₅ 7 = m/n, with m and n whole numbers.
7 is bigger than 1, so the log is positive.
m and n are positive.
Careful!
The log is a power, not a number times 5.
2 Work it out
A log is a power: log₅ 7 is the power of 5 that gives 7.
Raise both sides to the power n: 5ᵐ = 7ⁿ.
3 Find the contradiction
Prime factors: 5 to the m is only 5s, 7 to the n is only 7s.
A number has only one set of prime factors.
They can never be equal!
The opposite led to nonsense, so log₅ 7 is irrational.
4 Check
A calculator: log₅ 7 is about 1.209062, the digits going on with no pattern.
The closest pair: 5 to the 6 is 15625, 7 to the 5 is 16807.
Close, but never equal.
Right!
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.