Description: The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of Cohen p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007)
Ref | Expression | ||
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Assertion | relogdiv | |- ( ( A e. RR+ /\ B e. RR+ ) -> ( log ` ( A / B ) ) = ( ( log ` A ) - ( log ` B ) ) ) |
Step | Hyp | Ref | Expression |
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1 | efsub | |- ( ( ( log ` A ) e. CC /\ ( log ` B ) e. CC ) -> ( exp ` ( ( log ` A ) - ( log ` B ) ) ) = ( ( exp ` ( log ` A ) ) / ( exp ` ( log ` B ) ) ) ) |
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2 | resubcl | |- ( ( ( log ` A ) e. RR /\ ( log ` B ) e. RR ) -> ( ( log ` A ) - ( log ` B ) ) e. RR ) |
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3 | 1 2 | relogoprlem | |- ( ( A e. RR+ /\ B e. RR+ ) -> ( log ` ( A / B ) ) = ( ( log ` A ) - ( log ` B ) ) ) |