Theorem relogef

Description: Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007)

	Ref	Expression
Assertion	relogef	$$⊢A\in ℝ\to \log {\mathrm{e}}^A=A$$

Proof

Step	Hyp	Ref	Expression
1		relogrn	$$⊢A\in ℝ\to A\in \mathrm{ran}log$$
2		logef	$$⊢A\in \mathrm{ran}log\to \log {\mathrm{e}}^A=A$$
3	1 2	syl	$$⊢A\in ℝ\to \log {\mathrm{e}}^A=A$$