Theorem relogeftb

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

	Ref	Expression
Assertion	relogeftb	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( ( log ‘ 𝐴 ) = 𝐵 ↔ ( exp ‘ 𝐵 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		rpcnne0	⊢ ( 𝐴 ∈ ℝ+ → ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) )
2		relogrn	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ran log )
3		logeftb	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ran log ) → ( ( log ‘ 𝐴 ) = 𝐵 ↔ ( exp ‘ 𝐵 ) = 𝐴 ) )
4	3	3expa	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ran log ) → ( ( log ‘ 𝐴 ) = 𝐵 ↔ ( exp ‘ 𝐵 ) = 𝐴 ) )
5	1 2 4	syl2an	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( ( log ‘ 𝐴 ) = 𝐵 ↔ ( exp ‘ 𝐵 ) = 𝐴 ) )