Proving theorems and special cases (Part 13): Uniqueness of logarithms

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no. In this series of posts, we’ve seen that a conjecture could be true for the first 40 cases or even the first 10^{316} cases yet not always be true. We’ve also explored the computational evidence for various unsolved problems in mathematics, noting that even this very strong computational evidence, by itself, does not provide a proof for all possible cases.

However, there are plenty of examples in mathematics where it is possible to prove a theorem by first proving a special case of the theorem. For the remainder of this series, I’d like to list, in no particular order, some common theorems used in secondary mathematics which are typically proved by first proving a special case.

The next theorem is needed in calculus to show that \ln x = \displaystyle \int_1^x \frac{dt}{t}.

4. Theorem. Let a \in \mathbb{R}^+ \setminus \{1\}. Suppose that f: \mathbb{R}^+ \rightarrow \mathbb{R} has the following four properties:

  1. f(1) = 0
  2. f(a) = 1
  3. f(xy) = f(x) + f(y) for all x, y \in \mathbb{R}^+
  4. f is continuous

Then f(x) = \log_a x for all x \in \mathbb{R}^+.

In other blog posts, I went through the full proof of this theorem, which is divided — actually, scaffolded — into cases:

Case 1. f(x) = \log_a x if x is a positive integer.

Case 2. f(x) = \log_a x if x is a positive rational number.

Case 3. f(x) = \log_a x if x is a negative rational number.

Case 4. f(x) = \log_a x if x is a real number.

Clearly, Case 1 is a subset of Case 2, and Case 3 is a subset of Case 4. Once again, a special case of a theorem is used to prove the full theorem.

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  1. Proving theorems and special cases: Index | Mean Green Math

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