Theorem relogmul

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 Steve Rodriguez, 25-Nov-2007)

	Ref	Expression
Assertion	relogmul	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( log ‘ ( 𝐴 · 𝐵 ) ) = ( ( log ‘ 𝐴 ) + ( log ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		efadd	⊢ ( ( ( log ‘ 𝐴 ) ∈ ℂ ∧ ( log ‘ 𝐵 ) ∈ ℂ ) → ( exp ‘ ( ( log ‘ 𝐴 ) + ( log ‘ 𝐵 ) ) ) = ( ( exp ‘ ( log ‘ 𝐴 ) ) · ( exp ‘ ( log ‘ 𝐵 ) ) ) )
2		readdcl	⊢ ( ( ( log ‘ 𝐴 ) ∈ ℝ ∧ ( log ‘ 𝐵 ) ∈ ℝ ) → ( ( log ‘ 𝐴 ) + ( log ‘ 𝐵 ) ) ∈ ℝ )
3	1 2	relogoprlem	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( log ‘ ( 𝐴 · 𝐵 ) ) = ( ( log ‘ 𝐴 ) + ( log ‘ 𝐵 ) ) )