Theorem relogmuld

Description: The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 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	relogmuld	⊢ ( 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		relogmul	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( log ‘ ( 𝐴 · 𝐵 ) ) = ( ( log ‘ 𝐴 ) + ( log ‘ 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( log ‘ ( 𝐴 · 𝐵 ) ) = ( ( log ‘ 𝐴 ) + ( log ‘ 𝐵 ) ) )