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Metamath Proof Explorer

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 ) ) )