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Metamath Proof Explorer

Theorem relogdivd

Description: The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of Cohen p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses relogcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
relogmuld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion relogdivd ( 𝜑 → ( log ‘ ( 𝐴 / 𝐵 ) ) = ( ( log ‘ 𝐴 ) − ( log ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 relogcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2 relogmuld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3 relogdiv ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( log ‘ ( 𝐴 / 𝐵 ) ) = ( ( log ‘ 𝐴 ) − ( log ‘ 𝐵 ) ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( log ‘ ( 𝐴 / 𝐵 ) ) = ( ( log ‘ 𝐴 ) − ( log ‘ 𝐵 ) ) )