Theorem logcxpd

Description: Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	rpcxpcld.1	|- ( ph -> A e. RR+ )
	rpcxpcld.2	|- ( ph -> B e. RR )
Assertion	logcxpd	|- ( ph -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	|- ( ph -> A e. RR+ )
2		rpcxpcld.2	|- ( ph -> B e. RR )
3		logcxp	|- ( ( A e. RR+ /\ B e. RR ) -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )