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	|- ( ph -> A e. RR+ )
	relogmuld.2	|- ( ph -> B e. RR+ )
Assertion	relogdivd	|- ( ph -> ( log ` ( A / B ) ) = ( ( log ` A ) - ( log ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		relogcld.1	|- ( ph -> A e. RR+ )
2		relogmuld.2	|- ( ph -> B e. RR+ )
3		relogdiv	|- ( ( A e. RR+ /\ B e. RR+ ) -> ( log ` ( A / B ) ) = ( ( log ` A ) - ( log ` B ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( log ` ( A / B ) ) = ( ( log ` A ) - ( log ` B ) ) )