Metamath Proof Explorer

Theorem relogmul

Description: The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of Cohen p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007)

		Ref	Expression
	Assertion	relogmul	\|- ( ( A e. RR+ /\ B e. RR+ ) -> ( log ` ( A x. B ) ) = ( ( log ` A ) + ( log ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		efadd	\|- ( ( ( log ` A ) e. CC /\ ( log ` B ) e. CC ) -> ( exp ` ( ( log ` A ) + ( log ` B ) ) ) = ( ( exp ` ( log ` A ) ) x. ( exp ` ( log ` B ) ) ) )
2		readdcl	\|- ( ( ( log ` A ) e. RR /\ ( log ` B ) e. RR ) -> ( ( log ` A ) + ( log ` B ) ) e. RR )
3	1 2	relogoprlem	\|- ( ( A e. RR+ /\ B e. RR+ ) -> ( log ` ( A x. B ) ) = ( ( log ` A ) + ( log ` B ) ) )

Step

Hyp

Ref

Expression

efadd

 |-  ( ( ( log ` A ) e. CC /\ ( log ` B ) e. CC ) -> ( exp ` ( ( log ` A ) + ( log ` B ) ) ) = ( ( exp ` ( log ` A ) ) x. ( exp ` ( log ` B ) ) ) )

readdcl

 |-  ( ( ( log ` A ) e. RR /\ ( log ` B ) e. RR ) -> ( ( log ` A ) + ( log ` B ) ) e. RR )

1 2

relogoprlem

 |-  ( ( A e. RR+ /\ B e. RR+ ) -> ( log ` ( A x. B ) ) = ( ( log ` A ) + ( log ` B ) ) )