Theorem logcxpd

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

	Ref	Expression
Hypotheses	rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	rpcxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	logcxpd	⊢ ( 𝜑 → ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpcxpcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpcxpcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		logcxp	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( log ‘ ( 𝐴 ↑𝑐 𝐵 ) ) = ( 𝐵 · ( log ‘ 𝐴 ) ) )