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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 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 B = B ⁢ log ⁡ A
4 1 2 3 syl2anc ⊢ φ → log ⁡ A B = B ⁢ log ⁡ A