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	$$⊢\phi \to A\in {ℝ}^{+}$$
	relogmuld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
Assertion	relogdivd	$$⊢\phi \to \log \left(\frac{A}{B}\right)=\log A-\log B$$

Proof

Step	Hyp	Ref	Expression
1		relogcld.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		relogmuld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3		relogdiv	$$⊢\left(A\in {ℝ}^{+}\wedge B\in {ℝ}^{+}\right)\to \log \left(\frac{A}{B}\right)=\log A-\log B$$
4	1 2 3	syl2anc	$$⊢\phi \to \log \left(\frac{A}{B}\right)=\log A-\log B$$