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 ∈ ℝ → ( log ‘ ( exp ‘ 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		relogrn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ran log )
2		logef	⊢ ( 𝐴 ∈ ran log → ( log ‘ ( exp ‘ 𝐴 ) ) = 𝐴 )
3	1 2	syl	⊢ ( 𝐴 ∈ ℝ → ( log ‘ ( exp ‘ 𝐴 ) ) = 𝐴 )