Theorem relogeftb

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

	Ref	Expression
Assertion	relogeftb	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to \left(\log A=B\leftrightarrow {\mathrm{e}}^B=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpcnne0	$$⊢A\in {ℝ}^{+}\to \left(A\in ℂ\wedge A\ne 0\right)$$
2		relogrn	$$⊢B\in ℝ\to B\in \mathrm{ran}log$$
3		logeftb	$$⊢\left(A\in ℂ\wedge A\ne 0\wedge B\in \mathrm{ran}log\right)\to \left(\log A=B\leftrightarrow {\mathrm{e}}^B=A\right)$$
4	3	3expa	$$⊢\left(\left(A\in ℂ\wedge A\ne 0\right)\wedge B\in \mathrm{ran}log\right)\to \left(\log A=B\leftrightarrow {\mathrm{e}}^B=A\right)$$
5	1 2 4	syl2an	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to \left(\log A=B\leftrightarrow {\mathrm{e}}^B=A\right)$$