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

Proof

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