Description: Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007)
Ref | Expression | ||
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Assertion | relogeftb | |- ( ( A e. RR+ /\ B e. RR ) -> ( ( log ` A ) = B <-> ( exp ` B ) = A ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpcnne0 | |- ( A e. RR+ -> ( A e. CC /\ A =/= 0 ) ) |
|
2 | relogrn | |- ( B e. RR -> B e. ran log ) |
|
3 | logeftb | |- ( ( A e. CC /\ A =/= 0 /\ B e. ran log ) -> ( ( log ` A ) = B <-> ( exp ` B ) = A ) ) |
|
4 | 3 | 3expa | |- ( ( ( A e. CC /\ A =/= 0 ) /\ B e. ran log ) -> ( ( log ` A ) = B <-> ( exp ` B ) = A ) ) |
5 | 1 2 4 | syl2an | |- ( ( A e. RR+ /\ B e. RR ) -> ( ( log ` A ) = B <-> ( exp ` B ) = A ) ) |