Definition D1
For a natural number n, aⁿ is a multiplied n times.
Proof.
(1)a¹ = a, and for a natural number n, aⁿ⁺¹ = aⁿ × a.convention
(2)The base c is fixed as a number greater than 1, and log means the logarithm with base c.convention
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Lemma L1
If a > 0, then for every natural number n, aⁿ > 0
Proof.
(1)For n = 1, a¹ = a > 0.D1, assumption
(2)Suppose aⁿ > 0.assumption
(3)aⁿ⁺¹ = aⁿ × a.D1
(4)A product of positives is positive, so aⁿ⁺¹ > 0.(2), (3), A3
(5)By induction, it holds for every natural number n.(1), (4), A2
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Theorem T1
If a > 0 and n is a natural number, then log(aⁿ) = n log a
Proof.
(1)For n = 1, a¹ = a, so log(a¹) = log a.D1
(2)1 × log a = log a, so it holds at n = 1.(1), A5
(3)Suppose log(aⁿ) = n log a.assumption
(4)aⁿ > 0 and a > 0.L1
(5)log(aⁿ × a) = log(aⁿ) + log a(4), A1
(6)aⁿ⁺¹ = aⁿ × a, so log(aⁿ⁺¹) = log(aⁿ) + log a(5), D1, E
(7) = n log a + log a(3), (6)
(8) = (n + 1) log a(7), A4, A5
(9)By induction, it holds for every natural number n.(2), (8), A2
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(1)1 × 1 = 1 and 1 > 0.A5, A13
(2)log(1 × 1) = log 1 + log 1(1), A1
(3)log 1 + log 1 = log 1(1), (2), E
(4)log 1 = 0(3), A11
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Theorem T2
If a > 0 and n is an integer, then log(aⁿ) = n log a
Proof.
(1)If n is a natural number, T1 gives it directly.T1
(2)Suppose n = 0. Then a⁰ = 1.A7
(3)log(a⁰) = log 1 = 0 and 0 × log a = 0.(2), L2, A6, E
(4)Therefore it holds at n = 0.(3)
(5)Suppose n < 0 and set m = −n. Then m is a natural number.assumption
(6)aⁿ = a⁻ᵐ = 1/aᵐ.(5), A8
(7)aᵐ × aⁿ = 1 with aᵐ > 0, aⁿ > 0.(6), A9, L1, A10
(8)log(aᵐ) + log(aⁿ) = log 1 = 0(7), A1, L2
(9)log(aⁿ) = −log(aᵐ)(8), A12
(10) = −m log a = n log a(5), (9), T1
(11)The three cases cover all integers, so it holds for every integer n.(1), (4), (10)
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Corollary C
If a > 0 and r is rational, then log(aʳ) = r log a
Proof.
(1)Write r = p/k (p an integer, k a natural number) and set b = aʳ.convention
(2)By the definition of rational exponents, bᵏ = aᵖ.(1), A14
(3)b is taken to be positive.convention
(4)log(bᵏ) = k log b.(3), T1
(5)log(aᵖ) = p log a.T2
(6)k log b = p log a(2), (4), (5), E
(7)k ≠ 0, so log b = (p/k) log a = r log a(1), (6), E
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