PROOF 100
Prove.
The argument of a logarithm must be positive
On the positive reals.
A way to read this proof
First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.
Definition D1
logₐ x denotes the y satisfying aʸ = x.
Proof.
(1)The logarithm is a name asking the exponent in reverse: to what power must the base be raised to give x?convention
(2)For this name to have meaning, such a y must exist and be unique.convention
(3)If none exists, there is no value to refer to; if there are two or more, which one is meant is undetermined.convention
∎
Lemma L1
If a > 0 and y is real, then aʸ > 0
Proof.
(1)y + (−y) = 0.A8
(2)aʸ × a⁻ʸ = a⁰ = 1.(1), A1, A2, E
(3)If aʸ = 0, then 0 × a⁻ʸ = 0, contradicting (2).(2), A3, A9, A5
(4)Therefore aʸ ≠ 0, and by the same argument aᵘ ≠ 0 for every real u.(3)
(5)Setting u = y/2 gives y = u + u.convention
(6)aʸ = aᵘ⁺ᵘ = aᵘ × aᵘ.(5), A1, E
(7)aᵘ ≠ 0, so aᵘ × aᵘ > 0.(4), A4
(8)Therefore aʸ > 0.(6), (7), E
∎
Theorem T1
For a > 0, a ≠ 1, if x ≤ 0 then logₐ x is undefined.
Proof.
(1)Suppose x ≤ 0.assumption
(2)For every real y, aʸ > 0.L1
(3)If aʸ = x, then x > 0, contradicting (1).(1), (2), A5
(4)Therefore no y satisfies aʸ = x.(3)
(5)There is no value to refer to, so logₐ x has no meaning.(4), D1
∎
Theorem T2
The base of a logarithm needs both a > 0 and a ≠ 1.
Proof.
(1)Suppose a = 1.assumption
(2)For every real y, 1ʸ = 1.A6
(3)If x = 1, infinitely many y satisfy the condition.(2), D1
(4)If x ≠ 1, no such y exists at all.(2), D1
(5)Either way the value is not uniquely determined, so base 1 cannot be used.(3), (4), D1
(6)Now set a = −1 and let u = 1/2.assumption
(7)If aᵘ were real, then aᵘ × aᵘ = a¹ = −1.(6), A1, A7
(8)If aᵘ ≠ 0, then aᵘ × aᵘ > 0; if aᵘ = 0, then aᵘ × aᵘ = 0.A4, A3
(9)−1 < 0, so both cases contradict (7).(7), (8), A9
(10)Therefore, with a negative base, aᵘ leaves the real numbers.(9)
(11)The base needs both a > 0 and a ≠ 1.(5), (10)
∎
Remark
Can A4 be removed from this proof?
Being nonzero and being positive are different sentences. Find the line that crosses between them.